Work and Potential Energy

01 / UNDERSTAND & APPLY

1. Work tracks the component of force along displacement

For a particle, the work done by a force along a path is W = ∫F·dr. An infinitesimal contribution is F ds cosθ, where θ is the angle between force and the local displacement. Work is a signed scalar: positive when the force component helps motion, negative when it opposes motion, and zero when force is perpendicular to displacement at every point.

For a force constant in both magnitude and direction, the integral reduces to F·(r_final − r_initial). That special result depends only on the endpoint displacement. For a force whose direction follows the path, such as sliding friction on a stationary rough track, the work generally depends on path length instead. Do not replace a path integral by force magnitude times displacement magnitude without checking the angle and variation.

Work is measured in joules, with 1 J = 1 N m. Although torque also has dimensions of force times length, torque is not work: it involves a cross product rather than the dot product that selects displacement along force. A force–position graph gives work through signed area, while force–time area gives impulse.

A person can expend chemical energy while holding a stationary load, even though the mechanical work on the unmoving load is zero in the ground frame. This distinction is not a contradiction: muscular metabolism and microscopic internal processes are different energy transfers from the stated force acting through the load’s displacement. Name the object and frame before interpreting “work done.”

02 / UNDERSTAND & APPLY

2. Net work changes kinetic energy, whether or not the force is conservative

For a constant-mass particle in an inertial frame, Newton’s law gives F_net = ma. Along the displacement, a·dr = v·dv, so integrating gives W_net = (1/2)mv_final² − (1/2)mv_initial² = ΔK. The theorem does not require constant force, straight-line motion or conservation of mechanical energy. It requires summing the work of all forces acting on the particle.

Kinetic energy K = mv²/2 is nonnegative and depends on speed, whereas momentum is a vector and depends on velocity. A force can change momentum direction while doing no work, as the radial net force does in ideal uniform circular motion. Conversely, zero net work between two points tells you the speeds match, not that the velocity vectors match.

For one-dimensional position-dependent force, W = ∫F(x) dx. Negative regions of an F–x graph subtract from positive regions. If the computed final kinetic energy becomes negative before a proposed endpoint, the particle cannot reach that endpoint under the stated model: it turns around earlier or another assumption must change. Negative work is physical; negative kinetic energy is not.

Work and kinetic energy are frame dependent. A change to another inertial frame changes displacement and velocities, but the work–energy theorem remains consistent within that frame. In a non-inertial frame, appropriate inertial-force terms are needed; mixing a laboratory-frame force calculation with a moving-frame speed can create an apparent energy mismatch.

03 / UNDERSTAND & APPLY

3. Potential energy packages the work of conservative interactions

A force is conservative on the relevant connected domain when its work between two points is independent of the path, equivalently when work around every allowed closed loop is zero. Its work can then be written W_conservative = −ΔU. Potential energy belongs to the configuration of the interacting system, although we often abbreviate this by saying an object “has” gravitational or spring potential energy.

Near Earth’s surface with uniform g and upward coordinate y, U_g = mgy + C. Raising the object increases U, while gravity does negative work. An ideal Hooke-law spring with extension or compression x from its natural length has U_s = kx²/2 + C. Both positive extension and negative compression store positive spring energy relative to the unstrained choice of zero.

The additive constant C is arbitrary for a single fixed interaction model. Shifting the zero changes numerical U and total mechanical energy by the same constant but does not change forces, energy differences or motion. Setting gravitational potential to zero at the ground is convenient for near-surface problems; it is not a universal physical boundary condition.

In one dimension, F_x = −dU/dx. A potential increasing with x produces force toward decreasing x. In several dimensions the force is minus the gradient, F = −∇U. Knowing a potential’s value at one point does not determine force there; its spatial slope does. The same nonzero potential value can coexist with zero force at an extremum.

04 / UNDERSTAND & APPLY

4. Put each energy transfer in the equation once

If all relevant forces are conservative, ΔK + ΔU = 0 and mechanical energy E = K + U stays constant. If additional forces do work on the particle while the selected conservative interactions are represented by U, then Δ(K + U) = W_other. Do not include both gravity’s work and its potential-energy change on the same side as independent contributions; that counts the same interaction twice.

For a block sliding a distance s on a fixed horizontal rough surface with constant normal force N and kinetic-friction coefficient μ_k, friction does work −μ_kNs. In a simple block–surface energy budget, the corresponding mechanical-energy loss appears as internal thermal energy. The force opposes relative sliding; if the surface itself moves, work on the block and heat generation need not equal the same simple expression.

An applied force moving a load slowly against a conservative force does work approximately equal to the increase in potential energy when kinetic-energy change and dissipative losses are negligible. The word “slowly” is a dynamical assumption, not a reason to omit energy transfer. Raising a load at constant speed still needs positive input work even though net work and ΔK are zero.

Springs attached to moving supports, driven systems and explicitly time-dependent potentials require attention to energy supplied by the external driver. The simple constant mechanical-energy rule applies to an isolated, time-independent conservative model. When several objects or deformable bodies are involved, include the kinetic, elastic and internal energies of the chosen system rather than treating every centre-of-mass force–displacement product as the complete physical work.

05 / UNDERSTAND & APPLY

5. A potential graph shows force, speed and allowed motion

For one-dimensional conservative motion with total energy E, K(x) = E − U(x). Classically allowed positions satisfy U(x) ≤ E. A turning point with nonzero restoring force occurs where U = E and the speed momentarily vanishes before reversing. The graph’s vertical gap to the energy line determines speed, while the slope determines force; these are different geometric readings.

Equilibria satisfy dU/dx = 0. A strict local minimum is stable against sufficiently small displacements: the force pushes back toward the minimum. A local maximum is unstable. If the second derivative is positive at a stationary point, stability follows in the one-dimensional small-displacement test; a negative second derivative indicates instability. A zero second derivative is inconclusive and higher-order terms must be inspected.

For a harmonic well U = k(x − x₀)²/2, the turning positions are x₀ ± √(2E_relative/k), where E_relative is measured above the minimum. Speed is largest at x₀. The particle cannot cross an intervening region where U > E within classical mechanics, even if another allowed region exists beyond it. Quantum tunnelling is outside this classical chapter model.

A particle exactly at an unstable equilibrium with zero speed can remain there in the ideal equations; instability means a small perturbation grows, not that the unperturbed mathematical state has an unexplained initial force. At an equilibrium where U = E, avoid automatically calling the position an ordinary reflecting turning point: the local shape and initial conditions determine the motion.

06 / UNDERSTAND & APPLY

6. Energy alone cannot guarantee that contact is maintained

An ideal stationary smooth track exerts a normal force perpendicular to the instantaneous displacement, so its normal force does no work on a particle. Nevertheless, the force is essential to determine whether the assumed path is possible. An energy calculation may predict a speed at the top of a loop, but radial Newton’s law determines whether the track can supply the required inward force without pulling the particle away from the surface.

For a small particle moving on the inside of a smooth vertical circular loop of radius R, contact at the top requires N ≥ 0. The radial equation there is mg + N = mv_top²/R, so v_top² ≥ gR. If released from rest at height h above the bottom with no energy loss, mgh = mg(2R) + mv_top²/2. Combining the two conditions gives h ≥ 5R/2 for the limiting complete loop.

This familiar result has boundaries: the object is modeled as a sliding particle, not a rolling sphere with rotational energy, and the guide supplies only a nonnegative normal reaction on the inside. A bead threaded on a rigid wire or an object attached to a rigid rod has different constraint possibilities. A string cannot push, so it has its own tension-nonnegativity condition.

A normal force can do work when the supporting surface moves, and static friction can transfer energy in suitable moving-contact situations. Conversely, ideal pure rolling on stationary ground has zero instantaneous contact-point velocity and no work by ideal static friction on the rigid body at that contact. Always check actual point-of-application displacement rather than memorize “normal never works” or “friction always removes energy.”

07 / UNDERSTAND & APPLY

7. Power measures how rapidly work is transferred

Average power is work divided by the elapsed time, P_average = W/Δt. Instantaneous power delivered by a force is P = F·v, obtained by differentiating work. The SI unit is the watt, 1 W = 1 J s⁻¹. Two machines can perform the same work but have different powers because their completion times differ.

For a load lifted vertically at constant speed v with no other losses, the lifting force is mg and useful power is mgv. If it accelerates upward at a, the force is m(g + a) and its instantaneous mechanical power is m(g + a)v. At the instant v = 0, a finite force can have zero instantaneous mechanical power even though it immediately begins increasing kinetic energy.

With a constant delivered mechanical power P and negligible resistance in one-dimensional motion from rest, d(mv²/2)/dt = P gives v = √(2Pt/m). The implied driving force P/v diverges at v = 0, so an ideal constant-power model cannot describe a realistic finite-force start all the way to the first instant. It is an approximation once the drive is operating in its power-limited regime.

For speed-dependent resistance opposing motion, steady speed satisfies useful driving power equal to resistance magnitude times speed. If resistance is kv², the required power is kv³. Doubling speed then needs eight times the power in that model, not twice or four times. Other resistance laws or changing efficiency produce different scalings.

Efficiency η is useful output energy divided by input energy, or useful output power divided by input power for the same operating interval. An ideal energy-conserving passive device has η ≤ 1. If a pump raises water at volume flow rate Q through height h, ideal useful power is ρQgh; motor input is this quantity divided by efficiency. Keep volume flow, mass flow and total volume distinct.

08 / UNDERSTAND & APPLY

8. Original practice and model checks

Use an explicit energy reference and write the complete balance before solving. For power, identify the useful output and the time or flow rate over which it is supplied.

09 / UNDERSTAND & APPLY

9. Choose the shortest valid energy route

First decide whether you need net work, a conservative potential difference or a complete system-energy balance. Then choose endpoints that remove unnecessary unknowns, such as a turning point where speed vanishes. Retain dissipative work and rotational energy when they belong to the model. Use force equations separately for constraints, contact and any requested acceleration.

Check dimensions and signs at the end. Force times distance is energy, while force times speed is power. A negative work value can be correct; a negative squared speed means the assumed endpoint or motion is invalid. Efficiency cannot create missing energy. A constant offset in potential changes a label, not a physical prediction.

• Work uses displacement at the force application point and the force component along it.
• Net work changes kinetic energy even for nonconservative forces.
• Potential-energy change replaces conservative work; do not count both twice.
• Allowed motion lies where E ≥ U, with force given by minus the slope.
• Energy and constraint forces must both allow a proposed path.
• Power includes time or velocity; efficiency relates useful output to total input.