Work & Kinetic Energy Theorem Work & Kinetic Energy Theorem Push a cart and it speeds up; drag a sled and it slows down. What changed is not just the force, but how long and in which direction the force acted while the object moved. Work is a measure of energy transfer by a force through displacement. If a force helps the motion, it adds kinetic energy; if it opposes, it drains it. The kinetic energy theorem then gives a powerful shortcut: the total (net) work by all forces equals the change in kinetic energy. This single idea can replace many kinematics steps, especially when forces vary with position or when motion happens along curvy paths. It also reveals why some forces never change speed (like ideal constraints that act perpendicular to motion), and why friction turns ordered motion into heat, reducing mechanical energy. In everyday life you feel this when cycling uphill (your muscles do positive work against gravity, but gravity does negative work on you), or when braking a bike (brake friction does negative work, reducing kinetic energy that ends up as heat in the pads and rim). When forces vary with position, the work is the area under an F–x curve; with angles, the dot product decides the sign; and with time, power is the rate at which work is done. You will use these tools to move fluently between forces, energy, and motion. remember Big picture: Forces change speed only through the component along motion. Parallel component adds/subtracts kinetic energy (work), perpendicular component bends the path but does no work. Energy transferred by a force through displacement: dot product of force and displacement. Positive if the force component aids motion, negative if it opposes. Work Kinetic Energy Energy an object has due to its motion: K = 1 2 m v 2 for a particle of mass m and speed v . Power Rate of doing work or transferring energy. Instantaneous: P = F v . Average: P avg = W/ t . Conservative Force A force whose work between two points is path-independent and equals the negative change in potential energy (e.g., gravity, ideal spring). A force whose work depends on path and cannot be captured fully by potential energy (e.g., kinetic friction, drag). It often converts mechanical energy to heat. Non-conservative Force Work–Energy Theorem For a particle (or center of mass), net work done by all forces equals the change in kinetic energy: W net = K . Work uses a dot product: W = F s = F s , where is the angle between force and displacement. This instantly tells you the sign and magnitude: 0 = 1 (max positive), 90 = 0 (zero), 180 = -1 (max negative). Units of work and energy are joules ( J ). The dot product connects vector forces with scalar energy changes, making it a clean bridge from Newton’s laws to energy methods. Work by a constant force Dot product definition for constant force along a straight displacement. Use this formula only when the force acting on the object maintains a constant magnitude and direction throughout the displacement. Angle rules: 0° → maximum positive work; 90° → zero work; 180° → maximum negative work. Forces always perpendicular to velocity (ideal centripetal, ideal normal on smooth circular track) do zero work. tip For changing forces or curving paths, break motion into tiny displacements and add contributions. In the limit, work is the line integral of force along the actual path. In one dimension with force along x , it becomes the area under F(x) vs x . Always track the direction: if F x and dx have the same sign, the small work dW is positive; opposite signs make it negative. Work by a variable force General definition of work along the actual path. Graphically, the work done is the signed area under the force-displacement curve from the initial to the final point. Graphical meaning: In 1D, plot F x versus x . The signed area between the curve and the x -axis from x i to x f equals the work. A straight line through the origin (Hooke’s law) gives a triangle area. If the curve dips below the axis, that area counts negative—indicating the force opposes motion and reduces kinetic energy. Displacement x Area under the line equals the work stored in the spring. custom Linear Hooke’s law line: F = k x, area under curve is triangular. Force along x Unstretched spring k x Final extension control dependent 2D PLOT Spring force vs extension (Hooke’s law) F = k x Spring constant Ideal massless spring following Hooke’s law: F = kx (restoring). Quasi-static stretching so external force balances spring force in magnitude. External agent does positive work against the restoring force. Work done is stored as potential energy in the spring. Define zero potential at natural length. Elastic potential energy of a spring: U s = 1 2 k x 2 U s = 1 2 k x 2 Energy stored in an ideal spring compressed or stretched by x. Valid within elastic (linear) region. Kinetic energy quantifies motion: K = 1 2 m v 2 . A heavier or faster object stores more kinetic energy, and doubling speed quadruples K . The work–energy link says that whenever net forces push along the motion, K rises; when they oppose, K falls. Because energy is scalar, this method avoids component-by-component equations and shines in problems with multiple forces or curving paths. Translational kinetic energy of a particle or center of mass. Kinetic energy Quantifies the energy stored due to motion, simplifying calculations when multiple forces act or the path curves. Particle (or center of mass) motion with constant mass. Newton’s second law holds: F net = m a . W net = K Start from Newton’s second law. Use d dt ( 1 2 v 2) = v d v dt . Power by net force equals rate of change of kinetic energy. Integrate over time. Net work equals change in kinetic energy. Work–Energy Theorem: W net = K Applicability and limits: The work–energy theorem holds for a particle or the center of mass of a system with constant mass in the non-relativistic regime. Include all real forces (gravity, normal, tension, friction, spring, applied). If some forces are perpendicular to motion throughout (ideal constraints), they do zero work and can be ignored in the energy balance. Non-conservative forces are allowed; their work simply appears on the left-hand side and typically reduces K . Rate of doing work at an instant. Instantaneous power The power output is determined by the component of the force acting parallel to the object's instantaneous velocity. Average power: P avg = W/ t Definition of instantaneous power. Average value over the interval. P avg = W t Finite work W done uniformly or non-uniformly over time interval t > 0 . Average rate of work over a time interval. Distinct from instantaneous power P = F v . Average vs instantaneous power: A car engine might have high instantaneous power during acceleration, but its average power over a long trip is lower if it cruises steadily. Power depends on both force and velocity; even a small force can deliver high power if the object is moving very fast along the force direction. e = v 2 - v 1 u 1 - u 2 Newton’s law of restitution (empirical). Take care with signs and directions. e = 1 (elastic), e = 0 (perfectly inelastic). Coefficient of restitution: e = v 2 - v 1 u 1 - u 2 1D head-on collision or components along line of impact. Short impact, internal impulsive forces dominate during contact. Define velocities positive along the line of impact. Ratio of relative separation speed to relative approach speed along the line of impact. Direction signs matter. In restitution problems, define one positive direction and stick to it. Use u 1 - u 2 and v 2 - v 1 consistently along the line of impact. Mixing signs or taking magnitudes blindly leads to wrong e. neet-alert Strategy with the work–energy theorem: 1) Identify initial and final states for the particle/center of mass. 2) Compute work by each force between those states. Conservative forces can be handled via potential energy change; non-conservative via explicit work (often negative). 3) Sum to get W net . 4) Set W net = K and solve for the unknown (speed, distance, compression, etc.). m = 2.0 kg F = 10 N s = 5.0 m Angle = 60 Friction negligible u = 0 J; m/s Use W = F s and W net = K (only one force does work along motion). Work along displacement; dot product sign easy Only the parallel component contributes. Initial kinetic energy is zero. Compute final speed. Work W by the force; final speed v A 2.0 kg cart is pulled by a 10 N force at 60° to the horizontal over 5.0 m on a frictionless surface. Starting from rest, find the work by the force and the final speed. Interpretation: The force’s horizontal component is 5 N, so moving 5 m does 25 J of positive work, all of which becomes kinetic energy. Even though the force has a vertical component, it does no work vertically here because there is no vertical displacement. Area under the F–x curve. Compute speed from work–energy. A 2.0 kg block moves along x under a variable force F(x) = 5x + 2 (N), from x = 0 to x = 3.0 m. It starts from rest. Find the work done and the final speed. Work W; final speed v J; m/s m = 2.0 kg F(x) = 5x + 2 , N x i = 0, x f = 3.0 , m u = 0 medium Work by variable force; area under curve Use W = 0 3 (5x + 2) ,dx and then W = K . Graph check: F(x) rises linearly, so the area under it from 0 to 3 m is a trapezium with average height (F 0 + F 3)/2 = (2 + 17)/2 = 9.5 N and width 3 m, giving W = 9.5 3 = 28.5 J—same as the integral. m = 1.0 kg u = 4.0 m/s k = 400 N/m = 30 k = 0.20 g = 9.8 m/s 2 hard Use W net = K along the incline during compression distance x. Resistive works: gravity component and friction. Spring stores elastic energy. Incline with friction and spring; energy bookkeeping Initial kinetic energy. Gravity opposes upward motion along the incline. Friction magnitude along the incline. Friction does negative work. Energy stored at maximum compression. Spring’s work on the block is - U = -200 x 2 ; equivalently move +200 x 2 to the right as stored energy. Stops momentarily at max compression. Quadratic in x. Positive root gives physical compression. A 1.0 kg block slides up a rough incline at 30° and compresses a spring of constant k = 400 N/m placed along the incline. The coefficient of kinetic friction is 0.20. If its initial speed at the start of compression is 4.0 m/s, find the maximum compression x of the spring before the block momentarily stops. Maximum compression x Energy flow view: The initial kinetic energy is exactly spent against gravity and friction, with the remainder stored in the spring. If friction were zero, compression would be larger. If the slope were gentler, gravity would sap less energy and compression would increase. On rough inclines: Use N = mg for kinetic friction f k = k N along the path. Do not use mg directly in f k = k mg unless = 0 . Also, static friction in rolling motion can do zero work on the rolling body’s center of mass. neet-alert Mechanical work needs displacement. If you hold the bag stationary, s = 0 so the external mechanical work on the bag is zero, even though your muscles expend metabolic energy. Holding a heavy bag does a lot of mechanical work because it feels tiring. High power implies a large force. Power depends on both force and velocity along the force: P = F v . A small force at high speed can deliver large power; a large force at zero speed delivers zero instantaneous power. Situation Force Displacement direction Sign of work Reason Lifting a block up Gravity Upward Negative Force opposite to displacement Lowering a block down at constant speed Gravity Downward Positive Force along displacement; external agent does negative work Uniform circular motion Centripetal Tangential Zero Force ⟂ displacement at every instant Sliding with kinetic friction Friction Along motion Negative Opposes motion; dissipates energy as heat Pulling at 0° on smooth floor Applied force Along motion Positive Aids motion; increases K Pushing at 120° on smooth floor Applied force Along motion Negative Component opposes motion Particle or center-of-mass treatment; rotational KE not included unless stated. Mass is constant; no mass ejection or accumulation. Non-relativistic speeds; K = 1 2 m v 2 is valid. For conservative forces, you may use potential energies; for non-conservative, compute work explicitly. Integrals follow the actual path taken, not just endpoints (unless force is conservative). Assumptions and boundaries you must check A reliable plan for work–energy questions Mark the initial and final states clearly. List all forces that do work between these states; decide which are conservative. Compute W : dot product for constants, area under curve or integral for variables. Use W net = K ; or K i + U i + W nc = K f + U f . Solve for the unknown; sanity-check units and sign. Hyperbola: P avg = W / Δt for a fixed W. Doing the same work faster requires higher average power. custom Average power P avg for fixed W Time interval Δt control control Δt P avg dependent P avg = W 1 s mark Δt = 1 P avg = W/2 Halved power Δt = 2 Interpretation of the power hyperbola: If W is fixed, doubling the time halves P avg . Instantaneous power can vary wildly during a motion, but the area under the P(t) curve over an interval always equals the total work in that interval. COS for work sign: 0° → Cos positive (Adds energy); 90° → Orthogonal (Zero work); 180° → Subtracts (Negative work). Tie angle to sign via cosine. Conservative vs non-conservative in energy form: K i + U i + W nc = K f + U f . Here U groups conservative forces (like gravity, spring) via potential energies. W nc captures non-conservative works (friction, air drag, applied forces with no potential). This mixed form often simplifies multi-force problems and aligns directly with the work–energy theorem. Path independence: For conservative forces, W = - U depends only on endpoints. Any closed loop gives zero net work. For non-conservative forces, the work depends on the exact path taken. tip Zero work in constrained motion: In ideal uniform circular motion the centripetal force and ideal normal are always perpendicular to instantaneous displacement, so they do zero work. They change direction (and therefore momentum vector), but not speed (kinetic energy). Outside scope alerts: Variable mass systems (like rockets) need momentum analysis with mass flow and are not handled by K = 1 2 m v 2 with constant m. Similarly, relativistic speeds require K = ( - 1)mc 2 ; stick to non-relativistic problems here. Internal vs external work: The work–energy theorem in the form W net = K uses net external work on the system (particle/CM). Internal forces can redistribute energy within the system but do not change the center-of-mass kinetic energy unless they eject/ingest mass. Constraint forces: Normal and tension can sometimes do work (e.g., a rope pulling a sled does positive work). They do zero work when always perpendicular to motion (e.g., smooth bead on a fixed circular wire) or when the point of application doesn’t move along their direction. Component method reminder: In 2D/3D, either use F d r directly or project along the displacement direction: F = F . For integrals, parametrize the path and integrate along it. Consistent sign conventions are critical. Units and dimensions: Work and energy share units joule (J) and dimensions M L 2 T -2 . Power is watt (W) with dimensions M L 2 T -3 . 1 W = 1 J/s. In some contexts, 1 horsepower ≈ 746 W. NEET heuristics: If forces are messy but you only need speed or distance, try the work–energy route. If you need time or acceleration profile, kinematics or dynamics may be better. Always check if some forces do zero work—dropping them simplifies the algebra instantly. Energy transferred by a force through displacement: W = F d r . Work Kinetic Energy Energy of motion: K = 1 2 m v 2 . Work–Energy Theorem W net = K linking net work to change in kinetic energy. Rate of doing work: P = F v ; average P avg = W/ t . Power Conservative Force Path-independent work; admits a potential energy U . Coefficient of Restitution Collision elasticity measure: e = v 2 - v 1 u 1 - u 2 . U s = 1 2 k x 2 for an ideal spring. Elastic Potential (Spring) U s End-of-lesson recap