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Force

In physics, a force is a push or pull upon an object resulting from the object's interaction with another object, which can cause the object to accelerate, decelerate, or change direction.^[1] Forces always arise from interactions and cease when the interaction ends.^[1] A force is a vector quantity, characterized by both magnitude and direction, and is measured in the SI unit of the newton (N), defined as the force required to accelerate a mass of one kilogram at a rate of one meter per second squared.^[2] When multiple forces act on an object, their effects are determined by vector addition; balanced forces result in no net change in motion, while unbalanced forces produce acceleration proportional to the net force and inversely proportional to the object's mass, as described by Newton's second law (F = ma).^[3] The concept of force is foundational to classical mechanics, encapsulated in Isaac Newton's three laws of motion published in 1687.^[2] Newton's first law states that an object at rest remains at rest, and an object in uniform motion continues in a straight line unless acted upon by an unbalanced force, establishing the idea of inertia.^[3] The third law asserts that for every action, there is an equal and opposite reaction, meaning forces always occur in pairs.^[2] Forces are classified into contact forces, which require physical touch between objects (such as friction, tension, normal force, and applied force), and non-contact forces (or field forces), which act at a distance without direct interaction (such as gravitational, electric, and magnetic forces).^[1] At the most fundamental level, all forces in nature are manifestations of four interactions: the gravitational force (weakest, responsible for planetary motion and weight), the electromagnetic force (governing electricity, magnetism, and chemical bonds), the weak nuclear force (involved in radioactive decay), and the strong nuclear force (binding atomic nuclei).^[3] These principles extend beyond classical physics into relativity and quantum mechanics, where force concepts are reformulated but remain central to understanding physical phenomena.^[3]

Historical Development

Pre-Newtonian Concepts

In ancient Greek philosophy, Aristotle conceptualized force primarily as an efficient cause or "mover" that initiates and sustains motion in bodies, distinguishing between natural and unnatural types of movement. Natural motion arises from inherent qualities within an object, such as heaviness compelling earth and water to move downward toward their natural place at the center of the universe, or lightness driving air and fire upward.^[4]^[5] This view positioned force not as an abstract quantity but as a teleological agent aligned with an object's essence and purpose, where motion ceases once the body reaches its proper position unless interrupted.^[6] Aristotle further differentiated natural forces from violent or unnatural ones, which require an external agent to compel motion contrary to an object's inherent tendencies, such as throwing a stone upward against its heaviness.^[7]^[8] Violent motion was seen as temporary and dependent on continuous application of the mover, with the speed proportional to the force exerted and inversely to the resistance encountered, though without quantitative measurement.^[9] This qualitative framework influenced physics for centuries, emphasizing motion's directional and qualitative properties over precise dynamics. During the medieval period, scholars like Jean Buridan and Nicole Oresme advanced these ideas through the impetus theory, positing that a projected body acquires a temporary, force-like quality or "impetus" from the initial mover, which sustains motion until dissipated by external resistance such as air or gravity.^[10]^[11] Buridan described impetus as an impressed motive power that decreases over time, explaining continued projectile motion without ongoing external force, while Oresme extended this to falling bodies by suggesting impetus could accelerate descent if aligned with natural heaviness.^[12] These developments refined Aristotelian distinctions by introducing a quasi-conserved quality to violent motion, bridging natural tendencies and external influences in qualitative terms. Earlier, Archimedes applied mechanical principles to forces through his analysis of levers, demonstrating how balanced weights on a beam pivoted around a fulcrum could achieve equilibrium or amplify effort without invoking Aristotelian movers directly.^[13] In works like On the Equilibrium of Planes, he illustrated practical force applications, such as using longer arms to counterbalance heavier loads, providing empirical insights into mechanical advantage that prefigured later statics.^[14] These examples highlighted force as a relational property in rigid systems, influencing medieval engineers in constructing devices like catapults.^[15]

Newtonian Formulation

In the late 17th century, Isaac Newton synthesized earlier ideas on motion into a rigorous mathematical framework for understanding force, culminating in his seminal work Philosophiæ Naturalis Principia Mathematica, first published in 1687.^[16] This text marked a departure from qualitative philosophies, introducing force as a quantifiable entity central to dynamics and resolving longstanding debates in astronomy, particularly those surrounding planetary motion. Building briefly on pre-Newtonian concepts such as the medieval theory of impetus, which qualitatively explained sustained motion through an internal "impetus" acquired from an initial push, Newton shifted toward precise, mathematical descriptions.^[17] Newton's formulation drew key influences from contemporaries and predecessors, notably Galileo Galilei and René Descartes. Galileo's work on inertia, which posited that bodies in motion tend to continue uniformly unless acted upon, provided the foundation for Newton's concept of unaltered motion in the absence of external influences.^[18] In contrast, Newton rejected Descartes's vortex theory, which explained celestial motions through mechanical whirlpools of subtle matter, favoring instead action at a distance without such intermediaries.^[16] These influences allowed Newton to integrate terrestrial and celestial mechanics under a unified system. Central to Newton's approach was his definition of force as vis impressa, or impressed force: "An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of uniform motion in a right line. This force consists in the action only, and remains no longer in the body when the action is over."^[19] This conception framed force not as an enduring property but as a transient cause of motion change, enabling mathematical treatment. In the Principia, Newton applied this to resolve debates on planetary motion by deriving Johannes Kepler's empirical laws—such as elliptical orbits and the equal areas rule—from his theory of universal gravitation, demonstrating that an inverse-square attractive force between bodies accounts for these phenomena as approximations in a solar system with minimal perturbations.^[16]

Newtonian Mechanics

First Law of Motion

The first law of motion, also known as the law of inertia, states that every body perseveres in its state of rest or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.^[16] This principle establishes that objects maintain their velocity—whether zero (at rest) or constant in magnitude and direction—without the application of an external net force, introducing the concept of inertia as the inherent resistance of matter to changes in motion.^[20] Historically, this law overturned the Aristotelian view that objects naturally come to rest due to an intrinsic tendency toward immobility, positing instead that rest and uniform motion are equivalent states requiring no sustaining force.^[21] Aristotle's framework described motion as requiring continuous application of force, with natural motion for earthly bodies being toward rest at the center of the universe, a notion that dominated physics for nearly two millennia until challenged by Newtonian mechanics in the late 17th century. By contrast, Newton's formulation shifted the paradigm to one where change in motion demands an impressed force, laying the groundwork for modern dynamics.^[22] The law holds specifically in inertial reference frames, defined as those where an isolated object experiences no acceleration and thus maintains constant velocity, serving as the foundational condition for applying Newtonian mechanics.^[23] Newton invoked the concept of absolute space—immobile and independent of observable bodies—to justify this, as illustrated in his rotating bucket experiment: when a bucket filled with water is spun, the water's surface concaves due to centrifugal effects, demonstrating rotation relative to absolute space even if no external reference is visible, thereby evidencing true inertial motion.^[24] Conceptually, this implies that a net force of zero (\vec{F}_{\text{net}} = 0) results in zero acceleration (\vec{a} = 0), preserving uniform motion or rest.^[20]

Second Law of Motion

The second law of motion states that the net force acting on an object is equal to the product of its mass and its acceleration, expressed as the vector equation \vec{F}_{\text{net}} = m \vec{a}.^[25] This law quantifies how an unbalanced force causes a change in the object's velocity, building on the concept of inertia where no net force results in constant motion.^[26] In vector form, both force and acceleration are vectors, meaning the direction of the acceleration matches the direction of the net force, while the magnitude of the acceleration is inversely proportional to the mass for a given force.^[25] For instance, if multiple forces act on an object, their vector sum determines the net force and thus the resulting acceleration. The law derives from the rate of change of momentum, where momentum \vec{p} is defined as mass times velocity (\vec{p} = m \vec{v} for constant mass systems), so \vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}.^[26] When mass remains constant, this simplifies to \vec{F}_{\text{net}} = m \frac{d\vec{v}}{dt} = m \vec{a}.^[26] This formulation implies the SI unit of force, the newton (N), defined as the force required to accelerate a 1 kg mass by 1 m/s², or equivalently 1 N = 1 kg·m/s².^[3] A practical example is a constant force pushing a 10 kg cart horizontally across a frictionless surface, resulting in an acceleration of 2 m/s² in the direction of the push, as \vec{F}_{\text{net}} = 10 \, \text{[kg](/page/KG)} \times 2 \, \text{m/s}^2 = 20 \, \text{N}.^[27] Here, mass represents the object's resistance to acceleration (inertia), distinct from weight, which is the gravitational force mg where g \approx 9.81 \, \text{m/s}^2 on Earth.

Third Law of Motion

Newton's third law of motion states that for every action, there is always an equal and opposite reaction, or more precisely, the mutual actions of two bodies upon each other are always equal in magnitude and directed to contrary parts.^[28] This principle, articulated by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica (1687), implies that if body A exerts a force \mathbf{F}_{AB} on body B, then body B exerts an equal and opposite force \mathbf{F}_{BA} = -\mathbf{F}_{AB} on body A.^[29] These action-reaction pairs always occur simultaneously and act along the line connecting the two bodies, emphasizing the symmetry inherent in physical interactions.^[30] In analyzing systems of particles, the third law distinguishes between internal and external forces: internal forces are those action-reaction pairs between objects within the system, while external forces originate from outside the system.^[31] For an isolated system—free from external forces—the internal forces sum to zero because each pair cancels in magnitude and direction, resulting in no net force on the system as a whole.^[32] This has direct implications for momentum conservation: since the net external force is zero (referencing the second law's relation to net force), the total momentum of the isolated system remains constant over time.^[33] A classic example is the recoil of a firearm: when a bullet is fired forward by expanding gases, the gun experiences an equal backward force, propelling it in the opposite direction with momentum equal in magnitude but opposite to that of the bullet.^[34] Similarly, rocket propulsion relies on the third law, as the engine expels hot exhaust gases rearward at high velocity, generating an equal forward thrust on the rocket to achieve liftoff and acceleration in the vacuum of space.^[35] A common misconception is that action-reaction forces "cancel" each other out, leading to no motion; however, since these forces act on different bodies, they do not cancel for individual objects but instead cause each to accelerate according to its mass (as per the second law).^[36] For instance, in the gun recoil example, the forward force accelerates the lightweight bullet rapidly, while the equal backward force accelerates the heavier gun more slowly, producing noticeable kickback without violating the law.^[37]

Definition and Properties

In Newtonian mechanics, force is defined as a vector quantity that represents the interaction capable of producing a change in the motion of a body, characterized by its magnitude, direction, and point of application.^[38] This definition stems from Isaac Newton's second law, where force is proportional to the rate of change of momentum, but fundamentally, it quantifies the push or pull exerted on an object.^[39] The vector nature of force allows it to be represented in Cartesian coordinates as \vec{F} = (F_x, F_y, F_z), where each component corresponds to the force along the respective axis.^[40] Forces are broadly classified into two types: contact forces, which arise from direct physical interaction between objects, such as friction or tension, and field forces (also known as action-at-a-distance forces), which act through empty space without requiring contact, exemplified by gravitational or electrostatic forces.^[41] Contact forces fundamentally originate from electromagnetic interactions at the atomic level, while field forces are mediated by fields pervading space.^[38] A key property of forces in Newtonian mechanics is their additivity, governed by the superposition principle, which states that multiple forces acting on a body combine vectorially to yield the net force, such that \vec{F}_{net} = \sum \vec{F}_i.^[42] This principle ensures that the effect of concurrent forces is equivalent to the single resultant force obtained by vector summation.^[43] Historically, the concept of force evolved from Newton's notion of "impressed force" in his Philosophiæ Naturalis Principia Mathematica (1687), described as an action exerted on a body to alter its state of rest or uniform rectilinear motion, measured by the change in momentum it produces.^[18] This marked a shift from pre-Newtonian ideas like impetus to a quantitative, mathematical framework. In modern extensions, particularly special relativity, force is generalized to the four-force tensor f^\mu = \frac{d p^\mu}{d \tau}, where p^\mu is the four-momentum and \tau is proper time; in the low-velocity Newtonian limit, this reduces to the classical three-force \vec{F} = m \vec{a}.^[44]

Combining and Applying Forces

Force Equilibrium

Force equilibrium occurs when the vector sum of all forces acting on an object is zero, resulting in zero acceleration of the object's center of mass, as derived directly from Newton's second law of motion, \vec{F}_{net} = m \vec{a}, where if \sum \vec{F} = 0, then \vec{a} = 0. This condition holds for both objects at rest and those moving with constant velocity in the absence of net forces. To analyze force equilibrium, free-body diagrams (FBDs) are constructed, which isolate the object and depict all external forces acting on it as vectors, including their magnitudes and directions, while excluding internal forces or forces exerted by the object itself.^[45] These diagrams facilitate the application of equilibrium conditions by allowing the resolution of forces into components and verification that their sum is zero in each direction. Static equilibrium describes the case where an object remains at rest relative to an inertial frame, with \sum \vec{F} = 0 ensuring no translational motion and, briefly, no net torque for rotational stability (detailed in rotational dynamics). A classic example is a balanced scale, such as a beam balance where equal masses on either side produce equal and opposite gravitational forces and support reactions, keeping the beam horizontal without tipping.^[46] Dynamic equilibrium, in contrast, applies to objects moving at constant velocity, where \sum \vec{F} = 0 maintains uniform motion without acceleration, as seen in an idealized scenario of a hockey puck sliding on frictionless ice, opposed only by balanced air resistance or negligible forces.^[47] Equilibrium can be translational, focusing on zero net force for linear motion, or rotational, requiring zero net torque to prevent angular acceleration, though the latter is analyzed separately in torque discussions.^[48]

Net Force and Resultants

When multiple forces act on an object, their combined effect is described by the net force, or resultant force, which is the vector sum of all individual forces: \vec{R} = \sum_i \vec{F}_i. This resultant determines the overall motion of the object, as it alone produces the same acceleration as the system of forces would.^[49] Forces, being vectors, add according to the parallelogram law, first articulated by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica. In this method, two forces \vec{F_1} and \vec{F_2} are represented as adjacent sides of a parallelogram, with the resultant \vec{R} as the diagonal vector from the common origin. For more than two forces, the polygon law extends this principle: forces are drawn head-to-tail in a closed polygon if in equilibrium, or with the resultant as the vector from start to end point otherwise. To compute the resultant analytically, forces are resolved into components along perpendicular axes, such as Cartesian coordinates: R_x = \sum F_{ix}, R_y = \sum F_{iy}, and then |\vec{R}| = \sqrt{R_x^2 + R_y^2}, with direction given by \tan^{-1}(R_y / R_x).^[50]^[51]^[49] In applications like projectile motion, the net force is the vertical gravitational force mg downward (assuming no air resistance), while the horizontal component is zero; this unbalanced resultant causes constant horizontal velocity and downward acceleration of g = 9.8 \, \mathrm{m/s^2}. For an object on a frictionless inclined plane at angle \theta to the horizontal, the net force parallel to the plane is the component of gravity mg \sin \theta down the incline, leading to acceleration a = g \sin \theta along the surface.^[52]^[53] Graphical methods, such as force polygons, are particularly useful in engineering for visualizing resultants in concurrent force systems; each force is scaled and drawn sequentially head-to-tail, with the resultant as the straight-line vector closing the polygon. Unlike force equilibrium, where the resultant is zero and no acceleration occurs, a non-zero net force produces acceleration \vec{a} = \vec{R}/m in the direction of \vec{R}, as per Newton's second law.^[51]^[50]

Examples of Forces

Gravitational Force

The gravitational force is a fundamental attractive interaction between any two objects with mass, acting as a long-range, non-contact force that requires no medium for propagation. This force, often simply called gravity, is responsible for phenomena ranging from the falling of objects on Earth to the motion of celestial bodies. It is always attractive and directed along the line connecting the centers of the two masses, with no repulsive counterpart in classical mechanics. Isaac Newton formulated the law of universal gravitation in his 1687 work Philosophiæ Naturalis Principia Mathematica, stating that the magnitude of the gravitational force F_g between two point masses m_1 and m_2 separated by a distance r is given by

F_g = G \frac{m_1 m_2}{r^2},

where G is the gravitational constant. The force acts toward the center of each mass, and the law follows an inverse-square dependence on distance, meaning it weakens rapidly as separation increases. This formulation unified terrestrial and celestial mechanics, explaining why the same force causes apples to fall and planets to orbit. The value of G was first experimentally determined by Henry Cavendish in 1798 through a torsion balance apparatus that measured the subtle attraction between lead spheres, yielding G \approx 6.74 \times 10^{-11} \, \mathrm{m^3 \, kg^{-1} \, s^{-2}} (modern refinements confirm $6.67430 \times 10^{-11} with high precision). Cavendish's method isolated gravitational effects from other forces, providing the first quantitative link between mass and attraction without relying on astronomical observations. Applications of gravitational force include planetary orbits, where it provides the centripetal force necessary for stable elliptical paths as described by Newton's laws and Kepler's earlier empirical rules. Near Earth's surface, the force manifests as weight, defined as W = m g, where g is the local acceleration due to gravity, approximately $9.81 \, \mathrm{m/s^2}. This weight arises from Earth's mass pulling objects downward. In gravitational field representation, the force on a test mass m due to a larger body of mass M is F_g = m g, with g = G \frac{M}{r^2} pointing toward the body's center, treating the field as a vector quantity independent of the test mass. The inverse-square law assumes point masses or spherically symmetric distributions, where the force outside a uniform sphere equals that of a point mass at its center; deviations occur for non-spherical or extended bodies, such as tidal effects from irregular shapes. This limitation highlights the law's idealization for practical calculations in astrophysics and engineering.

Electromagnetic Force

The electromagnetic force is a fundamental interaction that acts between electrically charged particles, manifesting as both attractive and repulsive effects depending on the charges involved. It combines electric forces, which arise between stationary charges, and magnetic forces, which affect moving charges or currents. Unlike the always-attractive gravitational force, the electromagnetic force can produce both attraction and repulsion, and it is vastly stronger at short ranges, dominating atomic and molecular structures. This force is responsible for phenomena ranging from chemical bonding to the operation of electrical devices. The electric component of the force is governed by Coulomb's law, first experimentally established by Charles-Augustin de Coulomb in 1785 using a torsion balance to measure attractions and repulsions between charged objects. The law states that the magnitude of the electrostatic force F_e between two point charges q_1 and q_2 separated by a distance r is

F_e = k \frac{|q_1 q_2|}{r^2},

where k = \frac{1}{4\pi\epsilon_0} \approx 8.99 \times 10^9 \, \mathrm{N \cdot m^2 / C^2} is Coulomb's constant, with \epsilon_0 being the vacuum permittivity. The force is directed along the line joining the charges and is repulsive for like charges (both positive or both negative) and attractive for unlike charges, as demonstrated in Coulomb's memoirs where he verified the inverse-square dependence through precise measurements.^[54] For example, the electrostatic repulsion between two electrons separated by atomic distances contributes to the stability of electron shells in atoms. The electric force is mediated by the electric field \mathbf{E}, defined as the force per unit charge, which for a point charge is radial and falls off as $1/r^2.^[55] The magnetic component arises when charges are in motion, producing a force perpendicular to both the velocity and the magnetic field. This is captured in the magnetic force term \mathbf{F}_m = q (\mathbf{v} \times \mathbf{B}), where q is the charge, \mathbf{v} its velocity, and \mathbf{B} the magnetic field strength. The full electromagnetic force on a charged particle, known as the Lorentz force, combines both aspects:

\mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B}),

formulated by Hendrik Lorentz in his 1895 theory of electromagnetic phenomena in moving bodies to explain effects on charged particles in fields. The magnetic force does no work on the particle since it is always perpendicular to the velocity, but it can alter the direction of motion, as in the deflection of charged particles in magnetic fields. The magnetic field \mathbf{B} mediates this interaction, with lines of force indicating its direction and strength. A practical example is in electric motors, where the Lorentz force on current-carrying conductors in a magnetic field generates torque, causing rotation; for instance, in a simple DC motor, the interaction between the stator's field and the rotor's current produces continuous torque up to several newton-meters in typical designs.^[56]^[57] In the 1860s, James Clerk Maxwell unified these electric and magnetic forces into a coherent theory of electromagnetism by demonstrating that varying electric fields produce magnetic fields and vice versa, leading to the propagation of electromagnetic waves. His seminal 1865 paper introduced the concept of a pervasive electromagnetic field governed by four equations (now known as Maxwell's equations), which describe how \mathbf{E} and \mathbf{B} fields interact and mediate forces without direct contact between charges, analogous to the gravitational field but with dynamic wave-like properties. This unification revealed that light itself is an electromagnetic wave, with electric and magnetic fields oscillating perpendicular to the direction of propagation.^[58]^[59]

Contact Forces

Contact forces are those that arise from the physical interaction between two objects in direct contact, as opposed to action-at-a-distance forces like gravity or electromagnetism.^[60] In classical mechanics, these forces are essential for describing everyday phenomena such as pushing, pulling, or sliding objects.^[39] They typically act at the surface of contact and can be resolved into components perpendicular and parallel to that surface. The normal force is the component of the contact force perpendicular to the surface of interaction, acting to prevent objects from penetrating each other.^[61] For an object on a horizontal surface, the normal force equals the object's weight, balancing the gravitational force. On an inclined plane with angle θ, the normal force is given by N = mg \cos \theta, where m is the mass and g is the acceleration due to gravity.^[62] Friction is the tangential component of the contact force that opposes relative motion between surfaces. Static friction acts to prevent motion and has a maximum value of f_s \leq \mu_s N, where \mu_s is the coefficient of static friction.^[63] Once motion begins, kinetic friction takes over, with magnitude f_k = \mu_k N, where \mu_k is the coefficient of kinetic friction, typically less than \mu_s.^[63] Tension is the pulling force transmitted along a flexible connector, such as a rope or string, acting equally in all directions along its length.^[64] For a mass m hanging in equilibrium from a massless rope, the tension equals the weight, T = mg.^[64] Another example is the spring force, which arises from the elastic deformation of a spring and follows Hooke's law: F = -kx, where k is the spring constant and x is the displacement from equilibrium. This restoring force is directed toward the equilibrium position. In continuum mechanics, contact forces distributed over materials lead to stress, defined as force per unit area, \sigma = F/A, where A is the cross-sectional area. This concept quantifies internal forces in solids under load.^[65]

Derived Concepts

Torque and Rotational Force

Torque, often denoted as \vec{\tau}, is the rotational equivalent of force in linear motion, representing the tendency of a force to cause rotation about a pivot point or axis. It is defined as the cross product of the position vector \vec{r} from the axis to the point of force application and the force vector \vec{F}, given by \vec{\tau} = \vec{r} \times \vec{F}.^[66] The magnitude of torque is \tau = r F \sin \theta, where r is the distance from the axis to the line of action of the force (the lever arm), F is the force magnitude, and \theta is the angle between \vec{r} and \vec{F}.^[67] This formulation highlights that torque depends not only on the force but also on its perpendicular distance from the rotation axis, emphasizing the rotational effect over pure translation.^[68] In rotational equilibrium, the net torque on an object about a given axis must be zero, analogous to the condition for translational equilibrium where the net force is zero. This is expressed as \sum \vec{\tau} = 0, ensuring no angular acceleration occurs.^[69] For instance, on a balanced seesaw, the clockwise torque from one person's weight equals the counterclockwise torque from the other, maintaining equilibrium regardless of their linear positions as long as the moments balance about the fulcrum.^[70] The rotational analog of Newton's second law of motion relates net torque to angular acceleration \alpha through the moment of inertia I, which quantifies an object's resistance to rotational change based on its mass distribution. The equation is \vec{\tau}_{\text{net}} = I \vec{\alpha}, where I has units of kg·m².^[71] This law predicts that greater net torque produces larger angular acceleration for a given I, similar to how net force affects linear acceleration.^[72] A classic example is a lever, where applying a force at the end of a long arm (large r) generates substantial torque with minimal force, amplifying rotational effect around the pivot.^[73] Torque connects to linear force concepts through angular momentum \vec{L}, defined for a rigid body as \vec{L} = I \vec{\omega}, where \vec{\omega} is the angular velocity. The time derivative of angular momentum equals the net torque, \vec{\tau}_{\text{net}} = \frac{d\vec{L}}{dt}, linking rotational dynamics to changes in rotational motion.^[74] In a seesaw scenario, unequal torques alter the angular momentum, causing rotation until balance is restored or motion continues.^[75]

Potential Energy from Forces

In physics, a conservative force is defined as one for which the work done by the force on an object moving between two points depends only on the initial and final positions, independent of the path taken.^[76] This property allows the work to be associated with a scalar potential energy function U, such that the force is the negative gradient of this potential: \mathbf{F} = -\nabla U.^[77] Examples include gravitational and elastic forces, where energy can be stored and recovered without loss. The gravitational potential energy between two point masses m_1 and m_2 separated by distance r is given by U_g = -\frac{G m_1 m_2}{r}, where G is the gravitational constant.^[78] Near Earth's surface, for an object of mass m at height h above a reference level, this approximates to U_g = m g h, where g is the acceleration due to gravity.^[79] For elastic deformations, such as a spring obeying Hooke's law \mathbf{F} = -k \mathbf{x} (where k is the spring constant and \mathbf{x} is the displacement from equilibrium), the associated potential energy is U_s = \frac{1}{2} k x^2.^[80] This quadratic form arises from integrating the force over displacement, representing the energy stored in the deformed material. The work-energy theorem states that the net work done on an object equals the change in its kinetic energy: W = \Delta K_E.^[81] For conservative forces alone, the work done is W = -\Delta U, linking changes in potential energy directly to kinetic energy variations.^[82] In contrast, non-conservative forces like friction do not store energy reversibly; instead, they dissipate mechanical energy as heat or other forms, preventing full recovery.^[83] For instance, kinetic friction converts motion into thermal energy, reducing the system's mechanical energy.^[84]

Conservation Laws

Conservation of momentum arises directly from Newton's third law of motion, which states that for every action there is an equal and opposite reaction. Consider an isolated system of particles interacting via internal forces. The total momentum \mathbf{P} of the system is defined as the vector sum \mathbf{P} = \sum m_i \mathbf{v}_i, where m_i and \mathbf{v}_i are the mass and velocity of the i-th particle. The time derivative of the total momentum is \frac{d\mathbf{P}}{dt} = \sum \mathbf{F}_i, where \mathbf{F}_i is the net force on the i-th particle. By the third law, internal forces between particles come in equal and opposite pairs, so their contributions to \sum \mathbf{F}_i cancel out. For an isolated system with no external forces, \frac{d\mathbf{P}}{dt} = 0, implying that \mathbf{P} is constant.^[85] This conservation law applies to closed systems, such as collisions between particles where external influences are negligible. In elastic or inelastic collisions, the total momentum before and after the interaction remains the same, enabling predictions of post-collision velocities from initial conditions. For example, in a one-dimensional collision between two masses, the equation m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f} holds, where subscripts i and f denote initial and final states.^[86] Conservation of energy in mechanical systems follows from the properties of conservative forces, which are those where the work done depends only on initial and final positions, not the path taken. For such forces, a potential energy U can be defined such that the force is \mathbf{F} = -\nabla U. The work-energy theorem states that the change in kinetic energy KE = \frac{1}{2} m v^2 equals the negative change in potential energy, so \Delta KE + \Delta U = 0. Thus, the total mechanical energy E = KE + U remains constant for systems governed solely by conservative forces, like gravity or electrostatic interactions. A deeper connection between symmetries and conservation laws is provided by Noether's theorem, which asserts that every continuous symmetry of the laws of physics corresponds to a conserved quantity. Specifically, translational symmetry in space—meaning the laws are invariant under shifts in position—implies conservation of linear momentum. This theorem, derived from the invariance of the action integral in Lagrangian mechanics, unifies the origins of these principles.^[87] These conservation laws hold rigorously only for isolated systems in non-relativistic classical mechanics. In non-isolated systems, external forces alter the total momentum or energy. Relativistic effects, as in special relativity, modify the definitions—momentum becomes \mathbf{p} = \gamma [m](/page/M) \mathbf{v} where \gamma = (1 - v^2/c^2)^{-1/2}—but conservation persists in inertial frames for the four-momentum./04%3A_Dynamics/4.03%3A_Relativistic_Momentum)

Units and Measurement

SI Units

The SI unit of force is the newton (N), defined as the force required to accelerate a mass of one kilogram at a rate of one meter per second squared, expressed as $1 \, \mathrm{N} = 1 \, \mathrm{kg \cdot m/s^2}. This definition stems from Newton's second law of motion, establishing force as the product of mass and acceleration. The newton was named in honor of Sir Isaac Newton for his foundational work on mechanics and adopted as the SI unit in 1948 by the General Conference on Weights and Measures (CGPM). Prior to this, force units varied by system, but the newton's adoption standardized measurements globally. In other systems, the dyne (dyn) from the centimeter-gram-second (CGS) system equals $10^{-5} \, \mathrm{N}, while the pound-force (lbf) in the imperial system is approximately $4.44822 \, \mathrm{N}. These conversions facilitate interoperability, with $1 \, \mathrm{dyn} = 1 \, \mathrm{g \cdot cm/s^2} and $1 \, \mathrm{lbf} = 1 \, \mathrm{lb \cdot ft/s^2} (where 1 lb ≈ 0.453592 kg). Force is commonly measured using devices such as spring scales, which rely on Hooke's law to indicate tension or compression, and dynamometers, which quantify torque or linear force in engineering applications. These instruments are calibrated against SI standards to ensure accuracy. Following the 2019 revision of the SI, the kilogram is now defined exactly in terms of the Planck constant (h = 6.62607015 \times 10^{-34} \, \mathrm{J \cdot s}), enhancing the precision of force measurements by anchoring them to fundamental constants rather than artifacts. This change eliminates uncertainties from physical prototypes, improving traceability for the newton in metrology.

Dimensional Analysis

Dimensional analysis examines the fundamental units of physical quantities to ensure consistency in equations and derive relationships between variables without solving full differential equations. The dimension of force, derived from Newton's second law F = ma, is expressed as [F] = MLT^{-2}, where M represents mass, L length, and T time.^[88] This formulation highlights force as a product of inertial mass and acceleration, providing a basis for scaling physical laws across different systems. The Buckingham π theorem formalizes dimensional analysis by stating that any physical relationship involving n variables with k independent dimensions can be reduced to n - k dimensionless groups, or π terms. In fluid mechanics, this theorem applies to drag force F_d, which depends on fluid density \rho, velocity v, characteristic length (e.g., radius r), and kinematic viscosity \nu. The theorem yields the drag coefficient C_d = \frac{F_d}{\rho v^2 r^2} as one π term and the Reynolds number Re = \frac{r v}{\nu} as another, leading to C_d = f(Re), where f is an unknown function determined experimentally.^[89] For high Reynolds numbers, C_d approaches a constant, implying F_d \propto \rho v^2 A, with A as the cross-sectional area, which aids in predicting aerodynamic forces without full simulations. Scaling laws in mechanics emerge from dimensional analysis by comparing similar systems, where lengths, times, and masses are scaled by factors \lambda_L, \lambda_T, and \lambda_M. For geometrically similar structures, forces scale with the square of the linear dimension (F \propto L^2) due to stress being independent of size, while weights scale with volume (W \propto L^3). This results in a strength-to-weight ratio decreasing as L^{-1} or M^{-1/3}, explaining why larger animals require proportionally thicker bones to support their mass.^[90] Such ratios guide engineering designs, like model testing, by ensuring dynamic similarity through matched dimensionless numbers. A classic example is the simple pendulum, where the period T depends on bob mass m, string length l, and gravitational acceleration g. Assuming T = C m^\alpha l^\beta g^\gamma with C dimensionless, equating dimensions yields \alpha = 0, \beta = 1/2, and \gamma = -1/2, so T = C \sqrt{l/g}. The exponent \alpha = 0 demonstrates the period's independence from mass, as it cancels out in the underlying equations of motion.^[91] Despite its utility, dimensional analysis has limitations: it reveals only the form of relationships and dimensionless groups but cannot determine numerical constants or the exact functional form of f in expressions like \pi_1 = f(\pi_2, \dots). For instance, in the pendulum case, C = 2\pi requires solving the differential equation, not just dimensions. Additionally, it assumes all relevant variables are identified, and superfluous or incomplete sets can lead to overly complex or inaccurate results.^[92]

Modern Revisions

Relativistic Mechanics

In special relativity, the concept of force is revised to account for the effects of high velocities approaching the speed of light, c. The relativistic momentum \mathbf{p} of a particle with rest mass m and velocity \mathbf{v} is given by \mathbf{p} = \gamma m \mathbf{v}, where the Lorentz factor \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} with v = |\mathbf{v}|.^[93] This expression arises from the invariance of the spacetime interval and ensures that momentum transforms correctly under Lorentz transformations, preserving conservation laws in relativistic collisions.^[94] The relativistic force is defined as the rate of change of this momentum with respect to proper time \tau, the time measured by a clock moving with the particle: \mathbf{F} = \frac{d\mathbf{p}}{d\tau}.^[95] Here, \mathbf{F} is the four-force, a four-vector whose spatial components in the particle's instantaneous rest frame correspond to the three-force experienced by the particle. This differs from the Newtonian \mathbf{F} = m \mathbf{a}, as the effective inertia increases with velocity due to \gamma, leading to anisotropic acceleration: parallel to \mathbf{v}, the acceleration scales as \gamma^{-3} a, while perpendicular components scale as \gamma^{-2} a.^[96] In the low-velocity limit where v \ll c, \gamma \approx 1 + \frac{1}{2} \frac{v^2}{c^2}, so \mathbf{p} \approx m \mathbf{v} and \frac{d\tau}{dt} \approx 1, recovering the Newtonian form \mathbf{F} = m \mathbf{a}. This relativistic formulation is essential in applications like particle accelerators, where protons or electrons are accelerated to speeds exceeding 0.999c. For instance, in the Large Hadron Collider, the Lorentz force from superconducting magnets provides the centripetal force to maintain circular orbits, but the required magnetic field strength increases with \gamma to counteract the growing relativistic momentum, enabling energies up to several TeV without exceeding c.^[97] Such facilities demonstrate how classical mechanics fails at high speeds, as Newtonian predictions would underestimate the energy needed and predict unbounded velocities.^[98] In general relativity, the notion of force undergoes further revision: gravitational effects are not true forces but manifestations of spacetime curvature caused by mass-energy. Free particles follow geodesics—the shortest paths in curved spacetime—rather than deviating under a gravitational force, as described by the equivalence principle and the geodesic equation.^[99] This geometric interpretation eliminates the need for a gravitational force field, with apparent forces arising from non-geodesic motion in accelerated frames.

Quantum Mechanics

In quantum mechanics, force is not a directly observable quantity in the same deterministic sense as in classical mechanics, where it is defined as the time rate of change of momentum. Instead, the concept of force emerges through expectation values of observables, as encapsulated by the Ehrenfest theorem. This theorem states that the expectation value of the force acting on a particle, \langle \mathbf{F} \rangle = -\left\langle \nabla V(\mathbf{r}) \right\rangle, equals the mass times the second time derivative of the expectation value of position, \langle \mathbf{F} \rangle = m \frac{d^2 \langle \mathbf{r} \rangle}{dt^2}, linking quantum dynamics to classical equations of motion on average. The theorem demonstrates how quantum systems can approximate classical behavior for expectation values, particularly when wave packets are well-localized compared to variations in the potential V(\mathbf{r}). The Heisenberg uncertainty principle imposes fundamental limits on measuring force precisely, as force relates to changes in momentum over position. The principle asserts that the product of uncertainties in position and momentum satisfies \Delta x \Delta p \geq \hbar/2, preventing simultaneous knowledge of a particle's exact location and velocity, which in turn obscures exact force determination since \mathbf{F} = dp/dt requires tracking momentum evolution. This intrinsic indeterminacy means that attempts to measure force at quantum scales inevitably disturb the system, leading to probabilistic outcomes rather than definite trajectories. For instance, in atomic physics, interatomic forces arise from gradients of quantum potentials, such as the Coulomb potential in molecules, where the force on a nucleus is the negative gradient of the total potential energy surface derived from the electronic wave function. A striking example of quantum forces deviating from classical expectations is quantum tunneling, where particles traverse potential barriers that would be impenetrable classically. In alpha decay, for example, the strong nuclear force confines an alpha particle within a Coulomb barrier, but the particle's wave function allows a nonzero probability of tunneling through, effectively experiencing a force that permits escape without surmounting the barrier height. This probabilistic penetration highlights how quantum forces, derived from potential gradients, enable phenomena like nuclear fusion in stars, where tunneling overcomes electrostatic repulsion between protons. The time evolution of quantum states, governed by the Schrödinger equation i \hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi, incorporates force indirectly through the Hamiltonian \hat{H} = \frac{\hat{\mathbf{p}}^2}{2m} + V(\mathbf{r}), where the potential term relates to the force via \mathbf{F} = -\nabla V. Solutions to this equation yield wave functions \psi(\mathbf{r}, t) whose probability densities |\psi|^2 evolve under the influence of these potential-derived forces, often resulting in stationary states or superpositions that defy classical intuition. In the classical limit, quantum mechanics recovers Newtonian force laws through Bohr's correspondence principle, which requires that quantum predictions match classical results for large quantum numbers or macroscopic scales, such as highly excited atomic states where de Broglie wavelengths become negligible. This principle ensures continuity between quantum and classical descriptions, with the Ehrenfest theorem providing the mathematical bridge for force dynamics.

Field Theories

In classical field theory, forces are conceptualized as arising from spatial variations, or gradients, in underlying fields that permeate space. For example, the gravitational force on a test mass m is expressed as \mathbf{F} = -m \nabla \phi, where \phi is the scalar gravitational potential satisfying Poisson's equation \nabla^2 \phi = 4\pi G \rho, with G the gravitational constant and \rho the mass density. This formulation shifts the description from direct action-at-a-distance to a continuous field mediating the interaction, as developed in Newtonian gravity and extended to electrostatics with the electric potential.^[100]/02%3A_Review_of_Newtonian_Mechanics/2.14%3A_Newtons_Law_of_Gravitation) Quantum field theory (QFT) reconciles quantum mechanics with special relativity by treating particles as quantized excitations of pervasive fields, while forces emerge from the exchange of virtual particles—transient, off-shell field disturbances that do not obey the usual energy-momentum relation. In quantum electrodynamics (QED), the electromagnetic force between charged particles is mediated by virtual photons, excitations of the photon field, enabling precise calculations of interactions like Coulomb scattering. This framework bridges classical field concepts with particle physics, where the fields are operator-valued and governed by commutation relations.^[101]^[102] The Lagrangian formalism underpins both classical and quantum field theories by providing a variational principle to derive equations of motion. The action S = \int \mathcal{L} \, d^4x is extremized, where the Lagrangian density \mathcal{L} typically takes the form of kinetic energy terms minus potential-like interactions, analogous to the non-relativistic L = T - V but generalized to relativistic invariants such as \mathcal{L} = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - V(\phi) for a scalar field. The Euler-Lagrange equations \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \right) - \frac{\partial \mathcal{L}}{\partial \phi} = 0 yield the field equations, facilitating quantization via path integrals.^[103]^[104] A key advancement in QFT is Yang-Mills theory, which extends gauge invariance to non-Abelian Lie groups, introducing self-interacting vector fields crucial for non-Abelian forces. Proposed for isotopic spin conservation, it describes the strong nuclear force in quantum chromodynamics (QCD) via SU(3) gauge fields, where gluons—massless, charged mediators—undergo nonlinear interactions due to the non-commuting group structure, leading to phenomena like asymptotic freedom.^[105]^[106] The Standard Model exemplifies QFT's unifying power, framing the electromagnetic, weak, and strong interactions within a single renormalizable gauge theory based on the group SU(3)_C \times SU(2)_L \times U(1)_Y, with fermions as chiral representations and Higgs mechanism for mass generation. This structure, developed through electroweak unification, accurately predicts scattering processes and particle masses, serving as the cornerstone of modern particle physics.^[107]

Fundamental Interactions

Strong and Weak Nuclear Forces

The strong nuclear force governs the interactions among quarks and is essential for the structure of hadrons and atomic nuclei. At the fundamental level, it binds quarks within protons, neutrons, and other hadrons through the exchange of gluons, which are massless vector bosons carrying color charge, as described by quantum chromodynamics (QCD). QCD, a non-Abelian gauge theory based on the SU(3) color group, was theoretically established in the early 1970s, with the key discovery of asymptotic freedom—where the strong coupling constant decreases at high energies—demonstrated by David Gross, Frank Wilczek, and Hugh Politzer. This property allows perturbative calculations at short distances while explaining quark confinement at larger scales.^[108] The strong force operates over an extremely short range of approximately $10^{-15} m, the scale of nuclear dimensions, beyond which it drops off rapidly due to color confinement. Within this range, it is roughly 100 times stronger than the electromagnetic force, enabling it to dominate and overcome the Coulomb repulsion between positively charged protons in the nucleus. The residual strong force between composite nucleons (protons and neutrons) arises from the exchange of virtual pions and is modeled by the Yukawa potential, which takes the approximate form V(r) \propto \frac{e^{-r}}{r} in natural units. This potential captures the attractive, short-ranged nature of the nuclear binding. Additionally, the strong interaction exhibits approximate isospin symmetry, an SU(2) invariance that treats up and down quarks (and thus protons and neutrons) nearly equivalently, owing to their small mass difference of about 3 MeV. This symmetry underlies the near-degeneracy in masses of mirror nuclei and simplifies models of nuclear structure.^[109]^[110]/08%3A_Symmetries_of_the_theory_of_strong_interactions/8.01%3A_The_First_Symmetry_-_Isospin) In atomic nuclei, the strong force provides the primary binding mechanism, with average binding energies per nucleon reaching up to 8-9 MeV near iron-56, as revealed by the nuclear binding energy curve. This curve, derived from mass defect measurements, shows that lighter nuclei gain stability through fusion (releasing energy by moving toward higher binding per nucleon), while heavier nuclei release energy via fission (splitting into fragments with higher average binding). For instance, the fusion of hydrogen into helium in stars or the fission of uranium-235 in reactors both exploit these strong-force-mediated bindings to liberate vast amounts of energy, far exceeding chemical reactions./University_Physics_III_-Optics_and_Modern_Physics(OpenStax)/10%3A__Nuclear_Physics/10.03%3A_Nuclear_Binding_Energy) The weak nuclear force, in contrast, mediates flavor-changing processes at an even shorter range of about $10^{-18} m and is responsible for radioactive beta decay and other transformations that alter particle identities. It is carried by the massive W^\pm and Z^0 bosons, with masses around 80-91 GeV/c^2, which were experimentally confirmed at CERN in 1983. In beta-minus decay, for example, a down quark in a neutron emits a W^- boson, transforming into an up quark (yielding a proton) while the W^- decays into an electron and antineutrino; this charged-current interaction changes the particle's flavor and charge. The neutral-current process, mediated by the Z boson, allows flavor-preserving scattering but contributes to processes like neutrino interactions.^[109]^[111] A hallmark of the weak force is its violation of parity conservation, meaning it distinguishes between left- and right-handed particles, unlike the strong and electromagnetic forces. This was experimentally verified in the 1957 Wu experiment, where beta electrons from the decay of polarized ^{60}Co nuclei were observed to be emitted preferentially opposite to the nuclear spin direction, indicating a preferred handedness in the weak current. The result supported the vector-axial vector (V-A) theory of weak interactions proposed by Feynman and Gell-Mann, resolving puzzles in beta decay spectra and paving the way for the electroweak unification in the Standard Model. The weak force's role in flavor changes also drives stellar nucleosynthesis, such as the conversion of protons to neutrons in the rapid proton capture process.^[112]

Unification Attempts

One of the earliest successful attempts at unifying fundamental forces was the electroweak theory, which merges the electromagnetic and weak nuclear forces into a single framework. Proposed independently by Steven Weinberg in 1967 and Abdus Salam in 1968, building on Sheldon Glashow's 1961 model, this theory posits that at high energies, the SU(2) × U(1) gauge symmetry governs both interactions, with the Higgs mechanism breaking the symmetry to yield the observed forces. Experimental confirmation came in 1973 through the discovery of weak neutral currents by the Gargamelle collaboration at CERN, which observed neutrino interactions without charged lepton production, aligning with the theory's predictions.^[113] This unification, formalized in the Glashow-Weinberg-Salam model, forms a cornerstone of the Standard Model and earned its progenitors the 1979 Nobel Prize in Physics. Building on electroweak unification, Grand Unified Theories (GUTs) seek to incorporate the strong nuclear force, combining the electromagnetic, weak, and strong interactions under a single gauge group at energies around 10^16 GeV. The minimal GUT, SU(5), proposed by Howard Georgi and Sheldon Glashow in 1974, embeds the Standard Model's SU(3) × SU(2) × U(1) into SU(5), predicting that quarks and leptons unify in multiplets and leading to observable consequences like proton decay with a lifetime of about 10^31 to 10^34 years. An extension, the SO(10) model introduced by Georgi in 1975, uses the larger SO(10) group to naturally include right-handed neutrinos, enabling seesaw mechanisms for neutrino masses and further predicting proton decay modes. While these theories elegantly unify three forces and explain charge quantization, proton decay searches at detectors like Super-Kamiokande have set lower limits on lifetimes exceeding 10^34 years, constraining but not ruling out minimal GUTs. Efforts to unify all four fundamental forces, including gravity, have led to frameworks like string theory and its extension, M-theory. In string theory, developed in the 1970s and refined through the 1980s, fundamental particles and forces arise from vibrational modes of one-dimensional strings in 10 spacetime dimensions, with different vibrations corresponding to gravitons, photons, gluons, and other mediators. M-theory, conjectured by Edward Witten in 1995, unifies the five consistent superstring theories into an 11-dimensional framework incorporating membranes (branes), where forces manifest as low-energy approximations of higher-dimensional dynamics.00140-3.full) These approaches promise a quantum theory of gravity but remain untested, as their predictions emerge at the Planck scale of 10^19 GeV. Significant challenges persist in these unification attempts, particularly the inclusion of gravity, which resists quantization within standard field theory approaches, leading to non-renormalizable infinities in perturbative quantum gravity. The hierarchy problem exacerbates this, questioning why the weak force scale (around 246 GeV) is so much smaller than the Planck scale without fine-tuning, a issue that GUTs and string theory address through mechanisms like supersymmetry or extra dimensions but without definitive resolution. Moreover, while the Standard Model has triumphed in describing electromagnetic, weak, and strong interactions with extraordinary precision—predicting phenomena like the Higgs boson discovered in 2012—it falls short on gravity, neutrino masses, and the nature of dark matter, which constitutes about 27% of the universe's mass-energy yet evades Standard Model particles.^[114]^[115] Ongoing experiments at the LHC and neutrino observatories continue to probe these gaps, but a complete unification remains elusive.

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Torque - Front Matter
The torque exerted by a force F → acting on a rigid body about a given axis is , τ → = r → × F → , where r → is the perpendicular vector from the axis to the ...
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[PDF] Rotational Motion 3 - Duke Physics
Here is the general definition of the torque of a force about a given reference point: The vector r specifies the location, relative to the reference point, of ...
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[PDF] Chapter 10 Dynamics of Rotational Motion 1 Torque
The vector definition of the torque can be written as: ~τ = r × ~F. (the torque vector). How is the angle φ defined? It is the angle between r and ~F. Recall ...
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TUTORIAL : TORQUE AND ROTATIONAL EQUILIBRIUM
The sum of the torques must equal zero. All the torques must be calculated with reference to the same point designated as the fulcrum, and any point may be ...
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Rotational dynamics
torque = lever arm × force, ... While the seesaw is moving there is practically no torque on the seesaw, and it rotates with uniform angular velocity.
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10.7 Newton's Second Law for Rotation – University Physics Volume 1
If more than one torque acts on a rigid body about a fixed axis, then the sum of the torques equals the moment of inertia times the angular acceleration:.
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Newton's Second Law for Rotation - Physics
Newton's second law states that the angular acceleration is proportional to the net torque and inversely proportional to the moment of inertia. Newton's ...
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Rotational Motion - Galileo
A familiar example of a body having forces applied at different points is the seesaw. Let's say we have a seesaw carrying two children, one weighing 20 kg ...
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Angular momentum
The angular momentum L about one of the object's symmetry axis is the product of the object's moment of inertia I times its angular velocity ω about this ...
[74]
The Law of Conservation of Angular Momentum - Physics
L = Iω · Angular momentum is a vector, pointing in the direction of the angular velocity. · If there is no net torque acting on a system, the system's angular ...
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45. 7.4 Conservative Forces and Potential Energy - UH Pressbooks
A conservative force is one for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken. We can ...
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[PDF] Lecture D8 - Conservative Forces and Potential Energy
In multiple dimensions, the condition for a force field to be conservative is that it can be expressed as the gradient of a potential function. That is,. F = − ...
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Gravitational Potential Energy - HyperPhysics
where G is the gravitation constant, M is the mass of the attracting body, and r is the distance between their centers. This is the form for the gravitational ...
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7.3 Gravitational Potential Energy – College Physics
The equation Δ PE g = m g h applies for any path that has a change in height of h , not just when the mass is lifted straight up. (See Figure 2.) It is much ...
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Elastic Potential Energy - HyperPhysics
According to Hooke's law, the force required to stretch the spring will be directly proportional to the amount of stretch.
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7.4 Conservative Forces and Potential Energy - UCF Pressbooks
Use the work-energy theorem to show how having only conservative forces implies conservation of mechanical energy. ... The work-energy theorem states that the net ...
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[PDF] 8. Conservative Forces and Potential Energy
Therefore, we can define, for conservative forces, an associated potential energy that, for a given object, depends only on its location. In particular ...
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7.5 Nonconservative Forces – College Physics - UCF Pressbooks
Friction, for example, creates thermal energy that dissipates, removing energy from the system.
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Conservative and non-conservative force-fields - Richard Fitzpatrick
Hence, friction is an example of a non-conservative force, because it dissipates energy rather than storing it.
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9.3 Conservation of Linear Momentum - University Physics Volume 1
Sep 19, 2016 · Recall Newton's third law: When two objects of masses m 1 m 1 and m 2 m 2 interact (meaning that they apply forces on each other), the force ...
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10 Conservation of Momentum - Feynman Lectures - Caltech
So, here is another law, which might be stated: If two bodies have equal masses, as measured by equal velocities at one velocity, they will have equal masses ...
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[PDF] Invariant Variation Problems
Emmy Noether. M. A. Tavel's English translation of “Invariante Variationsprobleme,” Nachr. d. König. Gesellsch. d. Wiss. zu Göttingen, Math-phys. Klasse, 235 ...
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1.4 Dimensional Analysis – University Physics Volume 1
The dimension of any physical quantity expresses its dependence on the base quantities as a product of symbols (or powers of symbols) representing the base ...
[88]
[PDF] 5.5 Buckingham Pi theorem - MIT
May 13, 2010 · First, toward the low-Reynolds-number end, the drag coefficient increases. Second, for high Reynolds numbers, the drag coefficient stays.
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[PDF] Dimensional analysis and scaling laws - Galileo
We can “solve” certain physical problems with- out actually doing the detailed calculations needed for a complete solution.
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[PDF] 4 The Behavior of a Simple Pendulum and a Precision ...
While dimensional analysis cannot determine the value of the number C, it correctly predicts the form of the dependence of the period on quantities that have ...
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Summary of each segment:
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[PDF] 8.033 (F24): Lecture 08: Using 4-Momentum - MIT OpenCourseWare
E = γ(v)mc 2 , p = γ(v)mv ,. (8.26) where v is that body's 3-velocity. This tells us that if we know a body's relativistic energy and relativistic momentum ...
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[PDF] RELATIVISTIC ENERGY AND MOMENTUM - UT Physics
momentum p = γmv. Thus, the action S = −mc2. Z worldline dτ,. (90) the Lagrangian L = −mc2q1 − v2/c2,. (91) the momentum p = +mγv = mv p1 − (v/c)2 . (92).
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[PDF] The proper-force 3-vector - UMSL
The distinction between proper (i.e. cell-phone detectable) and geometric (i.e. connection-coefficient) forces allows one to use Newton's 3-vector laws in ...
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[PDF] Special Relativity
Oct 6, 2010 · The 4-force can be written in terms of the 3-force as. . F. f = d p/dτ = (d p/dt)(dt/dτ) = γ F. ☛ 4-force in terms of 3-force where is the ...Missing: four- limit
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[PDF] Lecture 1b - U.S. Particle Accelerator School
Example: Consider a light bulb hanging in a boxcar moving at relativistic velocity. How long does it take a light ray, moving directly down in the boxcar ...
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28.5 Relativistic Momentum – College Physics - UCF Pressbooks
Relativistic momentum is classical momentum multiplied by a relativistic factor, using rest mass and velocity relative to an observer.
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General Relativity - University of Pittsburgh
In relativity, both special and general, they are timelike geodesics. That is, they are curves of greatest proper time, which is the analog of the straight ...
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4. Gravitation - Lecture Notes on General Relativity - S. Carroll
We are now prepared to examine the physics of gravitation as described by general relativity. This subject falls naturally into two pieces.
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[PDF] Quantum Field Theory - DAMTP
where the field strength is defined by. Fµν = ∂µAν − ∂νAµ. (6.2). The equations of motion which follow from this Lagrangian are. ∂µ. ∂L. ∂(∂µAν). = −∂µFµν ...
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[1503.05007] The Evolution of Quantum Field Theory, From QED to ...
Mar 17, 2015 · Abstract page for arXiv paper 1503.05007: The Evolution of Quantum Field Theory, From QED to Grand Unification. ... Standard Model. Crucial for ...
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[PDF] 2. The Lagrangian Formalism - DAMTP
The equations of motion arising from L are m3 x = 2m3! ˙y + m3!2x. @V. @x m3 = 2m3! ˙x + m3!2y. @V. @y. (2.94). The full solutions to these equations are ...
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Lagrangian formalism for fields - Scholarpedia
Aug 30, 2010 · The Lagrangian formalism is one of the main tools of the description of the dynamics of a vast variety of physical systems.
[104]
Conservation of Isotopic Spin and Isotopic Gauge Invariance
The paper explores local isotopic spin rotations, leading to isotopic gauge invariance and a b field related to isotopic spin, similar to the electromagnetic ...
[105]
[PDF] quantum yang–Mills Theory - Clay Mathematics Institute
The non-abelian gauge theory of the strong force is called. Quantum Chromodynamics (QCD). The use of QCD to describe the strong force was motivated by a ...
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[PDF] Quantum field theory and the Standard Model
We give an introduction to the elements of quantum field theory, to the construction of the basic Lagrangian for a general gauge theory, and pro- ceed with the ...
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[PDF] Advanced information on the Nobel Prize in Physics, 5 October 2004
Oct 5, 2004 · The failure of the Yukawa field theory to describe the strong nuclear forces satisfactorily led the physics community to doubt the relevance of ...
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Four Fundamental Interaction
The residual strong force between two protons can be described by the exchange of a neutral pion. Note, the W± is not included as an exchange particle for the ...
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DOE Explains...The Weak Force - Department of Energy
In the Standard Model of particle physics, the weak force is carried by subatomic particles called W and Z bosons. By changing quarks to other types of quarks, ...
[111]
Experimental Test of Parity Conservation in Beta Decay | Phys. Rev.
Experimental test of parity conservation in beta decay. CS Wu, E. Ambler, RW Hayward, DD Hoppes, and RP Hudson. Columbia University, New York, New York.
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Gargamelle - CERN
The signature of a neutral current event was an isolated vertex from which only hadrons were produced. By July 1973 they had confirmed as many as 166 ...
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The Standard Model | CERN
The Standard Model explains how the basic building blocks of matter interact, governed by four fundamental forces.
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History of dark matter | Rev. Mod. Phys.
The standard model of modern cosmology is unthinkable without dark matter, although direct detections are still missing. A broad perspective of how dark ...