Potential Energy & Conservation Potential Energy & Conservation Potential energy is stored energy associated with position or configuration. You feel it whenever you lift a bag and sense it can fall back down, or when a compressed spring snaps back. The key idea is: for conservative forces (like gravity and ideal spring), the work done depends only on starting and ending positions, not on the path. That lets us define a scalar potential energy function U so that energy changes are easy to track. If only conservative forces act, the sum K + U stays constant as motion unfolds; kinetic energy K grows exactly when U drops, and vice versa. Real life also has non-conservative forces (friction, air drag) that convert mechanical energy into internal energy (heat, sound), so K + U is no longer constant. Still, a single equation can handle everything: the work by non-conservative forces equals the change in mechanical energy, W nc = Δ(K + U). Choosing a reference level where U = 0 is like choosing altitude zero for heights: it is arbitrary but must be used consistently. Near Earth’s surface, gravitational potential energy increases by mgh when you go up by height h. In a spring, potential energy grows like ( 1 2 kx 2 ) with extension x. At larger scales, gravity’s potential is negative with zero at infinity: U(r) = −GMm/r. Negative does not mean impossible; it simply reflects our chosen zero. With these tools, roller coasters, vertical circles, springs, and many NEET problems collapse into clean, one-line energy statements. Think of money accounts: kinetic energy is cash in hand; potential energy is money parked in a savings account. With only conservative forces, money just moves between accounts without loss. Friction is like a fee that leaves the banking system as heat. remember Potential Energy (U) Stored energy due to position or configuration relative to a chosen reference level. Conservative Force A force whose work is path-independent and can be written as the negative gradient of a potential energy, so that W conservative = −ΔU. A force (e.g., kinetic friction, air drag) whose work depends on path and generally dissipates mechanical energy as heat or sound. Non-conservative Force Sum of kinetic and potential energies, E mech = K + U. Mechanical Energy Reference Level An arbitrarily chosen level where U is set to zero; only differences in U have physical consequences. Measures spring stiffness; F spring = −kx and U spring = (1/2)kx 2 for ideal linear springs. Spring Constant (k) Path Independence Property of conservative forces: work depends only on initial and final points, not the route taken. Turning Point Position where kinetic energy momentarily becomes zero (all mechanical energy is potential). Why define potential energy? Because it absorbs the effect of conservative forces into a single scalar. Instead of carrying force vectors through every step, you can track energy flow. The rule is simple: if only conservative forces act, K + U is constant. If non-conservative forces do work W nc (sign included), then K 2 + U 2 = K 1 + U 1 + W nc. This single statement handles lifting, sliding, springs, roller coasters, and vertical circles. It rarely fails and is usually faster and safer in exams than summing forces and accelerations along changing directions. Definition via work Potential energy change equals negative of work by conservative forces. Force–potential link (1D) Force is the negative slope of U(x). The component of the conservative force is determined by the negative gradient of the potential energy function. Gravitational near Earth Choosing U = 0 at h = 0, valid for small height ranges where g is nearly constant. The change in potential energy due to gravity depends only on the height change and the object's mass. Gravitational potential (general) Zero at infinity; negative values closer to the mass reflect a bound state. This potential defines the work done by gravity, quantifying the energy stored by a mass at a specific distance from a large body. Zero at x = 0 (natural length). Spring potential energy With only conservative forces, W nc = 0 and K + U is conserved. Mechanical energy accounting Choosing the zero of U does not change physics. If you raise all U values by a constant C, then K adjusts by −C to keep K + U unchanged. NEET problems often set U = 0 at the ground (for mgh) or at infinity (for −GMm/r). Pick whichever makes numbers cleaner, but stick to it throughout the problem. When comparing two points, only ΔU matters, never the absolute values. Smart reference: For near-Earth problems, set U = 0 at the lowest point of motion so all heights have positive mgh. For gravitational orbits, set U(∞) = 0 to use U(r) = −GMm/r cleanly. tip U s = 1 2 kx 2 Hooke’s law holds: F s = −kx. The spring is massless and elastic (no hysteresis). Reference U = 0 at x = 0. U s = 1 2 kx 2 To slowly stretch, external force matches spring force in magnitude. Work done by you is stored as spring potential energy. Since U s (0) = 0 by reference choice. Elastic potential energy in an ideal spring grows quadratically with extension or compression; it is always non-negative. A powerful diagnostic is the graph of U versus position. Where U is flat, force is zero. Where U falls steeply, force is large in the +x direction (since F = −dU/dx). Turning points occur where total energy E equals U (so K = 0). Motion is allowed only where U(x) ≤ E. This visual method is quick when algebra gets messy. Equilibrium (U = 0) x (spring extension) custom control U(x) dependent Parabola opening upward with vertex at the origin, symmetric for compression (−x) and extension (+x). U (spring) U = 0.5 k x 2 2D PLOT Spring constant Spring potential energy vs extension U s = (1/2)kx 2: symmetric energy well for an ideal spring. -GMm/ R E Surface R E Reference zero custom r (distance from Earth center) dependent U(r) Distance r is in units of Earth’s radius; U → 0 as r → ∞. U g Monotonic curve approaching 0 from below as r increases; more negative close to Earth. Gravitational potential energy vs distance G·M·m 2D PLOT Ug = -C / r Ug Near Earth’s surface, where g ≈ constant, Δ U g = mgΔh. This is extremely reliable for height changes small compared to Earth’s radius. If heights get very large (rockets, satellites), you must switch to U(r) = −GMm/r; it predicts smaller gains in potential for the same vertical displacement at high altitude because g weakens with r. Sign trap: Work by gravity is W g = +mgΔh when moving down and W g = −mgΔh when moving up. But Δ U g = − W g , so Δ U g is negative when descending and positive when ascending. Many errors come from mixing these signs. neet-alert K 2 + U 2 = K 1 + U 1 + W nc K 2 + U 2 = K 1 + U 1 + W nc Newton’s second law holds. Forces split into conservative (from U) and non-conservative (e.g., friction). Work–energy theorem applies. Work–energy theorem. Split work by type. Definition of potential energy for conservative forces. Substitute W con. Rearrange. Energy accounting form. Friction’s work is W f = − f k d (kinetic friction opposing motion), so it lowers mechanical energy by f k d. If a surface has variable friction or rolling resistance, integrate along the path. Air drag often depends on speed; energy methods still work if you can integrate its work or if question asks for loss estimates over a segment. Conservative vs Non-Conservative Forces Force Type Definition Work in Closed Loop Examples Potential Energy Associated? Conservative forces are Path-less Travellers, while Non-conservative ones pay the Friction Toll at every step. Conservative Force Work done is independent of the path and depends only on initial and final positions. The work done in a closed loop is zero ( ∮ F d r = 0 ). Gravitational force, Electrostatic force, Magnetic force, Elastic spring force. Yes, Potential Energy ( U ) is defined such that W c = - U . Non-Conservative Force Work done depends on the actual path followed between the two positions. The work done in a closed loop is non-zero ( ∮ F d r 0 ). Frictional force, Viscous force ( 6 rv ), Air resistance, Tension in a string. No, energy is dissipated as heat, sound, or light; no U is associated. conservative vs non conservative forces Closed-loop confusion: Only for conservative forces is the work around any closed loop zero. Friction on a loop always removes energy; do not set it to zero. neet-alert Conservative → has a potential U; K + U conserved. “Con serves U”: Conservative forces serve you a U. If only conservative forces act, K + U stays constant. Vertical circle is a classic application. Use energy to relate speeds at different heights, then check the string tension using dynamics at key points. The minimum bottom speed to just complete the circle comes from combining energy conservation with the zero-tension condition at the top. This two-step method is faster and more reliable than constant acceleration formulas, which do not apply along curved paths with changing direction. v b, = 5gR Minimum speed at bottom for a light string mass m and radius R to just complete a vertical circle No air resistance or bearing friction. String is light and inextensible. Gravity is uniform with acceleration g. Take bottom as U = 0. Gain in potential over 2R height. Conservation of mechanical energy. Radial force balance at the top. Just-complete condition: tension non-negative. Insert the minimum admissible v t 2. Vertical circle trap: Many students set v top = 0 at the highest point. That breaks the contact (tension goes negative). The minimum condition is v top 2 = gR, not zero. neet-alert Power ties time into the story: it measures how fast work is done or energy is transferred. For constant power delivery over a time Δt, the total work is W = P avg Δt. While this chapter focuses on energy conservation, power questions often appear alongside: for instance, “How much power must a motor supply to lift a mass at steady speed?” Energy methods plus P = F·v or P avg = W/Δt wrap such problems quickly. P avg = W t Total work W is done uniformly over a time interval ( t > 0 ). Definition of power as rate of doing work. P avg = W/ t Instantaneous definition. Average over a finite interval. Average power is total work done divided by the time taken; useful when energy transfer is uniform over an interval. Near-Earth gravitational potential: Δ U g = mg h. Work by gravity is −Δ U g . Average power is P avg = W in/Δt with W in = Δ U g . J, W Easy: Lifting a bag and average power Δ U g , work by gravity, and average power input by the lifter (ignoring accelerations). Pattern: NCERT Exemplar-style Potential energy increase. Gravity does negative work when lifting. Average rate of energy input. easy Mass m = 2.0 kg Height gain h = 3.0 m Time taken ( t = 2.0 , s ) Take g = 9.8 m/s 2 In problems with friction, treat friction’s work as W nc and insert its value with the correct sign into the energy equation. If the friction coefficient and normal reaction are known, W f = −μ k N d along the path. Always link geometry (inclines, curves, heights) to potential energy changes before plugging numbers. m/s K 2 + U 2 = K 1 + U 1 + W nc with U referenced to the bottom (U 2 = 0). Height drop h = d $. Medium: Block sliding down a rough incline Speed at the bottom. medium Vertical drop. Potential decreases. Normal reaction on incline. Energy lost to friction. Mechanical energy gained as kinetic. Final speed. NEET-like incline with friction Mass m = 1.5 kg Incline angle ( = 30 ), length along plane d = 2.0 m Coefficient of kinetic friction ( k = 0.20 ) g = 9.8 m/s 2 Springs trade energy with motion cleanly. If a mass is launched by a compressed spring on a rough horizontal surface, the initial spring energy ( 1 2 kx 2 ) is gradually lost to friction ( k N ) until the block stops. Equating the spring energy to work lost gives the stopping distance immediately, without ever computing acceleration. Medium-Hard: Spring launch with friction Distance the block slides before coming to rest. Initial energy E i = (1/2)kx 2. Final kinetic and spring energies are zero; all initial energy lost to friction W fr = −μ k mg s. Spring constant k = 200 N/m Compression x = 0.10 m Block mass m = 0.50 kg Horizontal surface with ( k = 0.25 ) g = 9.8 m/s 2 Stored spring energy. Energy lost to friction (in J). Stopping distance. Energy loss with friction medium Hard: Minimum bottom speed for completing a vertical circle Minimum speed v b ,min at the bottom so that the string stays taut throughout. Use energy between bottom and top, then apply the zero-tension condition at the top. Combine to find v b ,min. m/s Mass m String length R Uniform gravity g No air resistance Vertical circle completion condition Bottom as zero potential. Energy relation. Just-contact condition. Minimum bottom speed. hard Energy diagrams also pinpoint equilibrium types. A point where dU/dx = 0 is equilibrium. If d 2U/dx 2 > 0 there, U has a local minimum and the equilibrium is stable (like the bottom of a bowl). If d 2U/dx 2 < 0, it is unstable (like the top of a hill). Small nudges around stable points cause oscillations whose frequency relates to curvature of U. Common reference choices Near-Earth: U = 0 at ground or lowest point of motion. Springs: U = 0 at natural length x = 0. Universal gravity: U(∞) = 0, so U(r) = −GMm/r. Arbitrary shifts U' = U + C do not affect dynamics; only ΔU matters. Pick a useful U = 0 reference. Write E 1 + W nc = E 2, where E = K + U. Compute ΔU from geometry (heights, extensions). Compute W nc (friction, drag) with correct sign. Solve for the required speed, height, extension, or distance. Recipe for energy problems It can. For gravity with U(∞) = 0, U(r) = −GMm/r is negative for any finite r. Only differences in U are physical. Potential energy can’t be negative. Path-independence means W depends only on endpoints. It is zero only for closed loops (same start and end). For gravity between two different heights, work is not zero. If work is path-independent, it must be zero. Friction ‘destroys’ energy completely. Friction converts mechanical energy into internal energy (heat, sound); total energy of the universe is conserved. Non-conservative forces can sometimes be turned off in parts of motion: for example, a bead sliding on a smooth (frictionless) section and then entering a rough patch. Apply energy conservation on the smooth segment, then include W nc only where it acts. Segment-wise energy accounting avoids mixing incompatible assumptions. In multi-spring systems, the potential energies add: U total = (1/2)k 1 x 1 2 + (1/2)k 2 x 2 2, with x 1, x 2 linked by geometry. For springs in series or parallel, you can first find equivalent k eq and then use U = (1/2)k eq x 2 $ for quick estimates. For variable gravity, potential difference between r 1 and r 2 is ΔU = −∫ r 1 r 2 (−GMm/ r 2 ) dr = GMm(1/r 2 − 1/r 1). This recovers mgh when r 2 − r 1 ≪ r 1 by Taylor expansion. NEET rarely needs the calculus detail, but recognizing the limit helps avoid using mgh at very large heights. Average power questions often appear as quick add-ons. If an elevator of mass M rises at constant speed v, tension T = Mg and instantaneous power P = T v = Mg v. If it takes time Δt to raise by height h = vΔt, the average power P avg equals Mg h/Δt, which matches P = Mg v for constant speed. This cross-check confirms consistency between energy and power views. Energy conservation plus small dynamic checks solve many multi-step problems: find speed by K–U, then find forces or tensions using that speed. Conversely, if a non-conservative process limits motion (like friction stopping a block), compute the energy lost to find the distance or time scales. Practice swapping between these two viewpoints quickly. Big picture: E mech changes only by the work of non-conservative forces. Conservative forces reshuffle energy between K and U; they never remove it from the mechanical account. remember When combining multiple conservative potentials, add them algebraically: U total = U g + U s + …. The force is then the sum of the corresponding conservative forces, or equivalently F = −dU total/dx$(1D). This superposition makes complex systems manageable, such as a mass-spring under gravity with an offset equilibrium. Superposition of potentials (1D) Potentials add; forces from each potential add too. This principle applies when multiple conservative forces act on a particle in one dimension, allowing the total potential energy and net for Careful with variable-height paths: even if the path zig-zags, for gravity the total ΔU depends only on the net vertical height change. That is the essence of path independence. So a long ramp and a vertical drop ending at the same level have identical ΔU, but friction losses can still differ because friction depends on path length. For kinetic friction with constant N over distance d along the path. Work by friction on a straight path This formula applies when an object moves along a straight path against kinetic friction, where the normal force (N) and the coefficient of Potential wells trap motion if E < U at the walls. In a spring well E = (1/2)kA 2, the mass oscillates between ±A with speed greatest where U is smallest. In gravitational wells, bounded orbits correspond to total energy below zero (E < 0). These qualitative statements save time by predicting motion without full solutions. Sometimes energy changes hide in thermal or sound channels. If a 2 kg cart moving at 3 m/s hits a bumper spring and emerges at 2 m/s, the missing kinetic energy did not vanish; part was stored temporarily as U s and the rest dissipated as heat and sound. Writing energy conservation for the cart + spring + surroundings keeps the accounting exact. Time-resolved link between dynamics and power flow. Instantaneous power from force and velocity Calculates the instantaneous rate at which a force does work on a moving object during energy transfer. At turning points, K = 0. Energy methods identify the maximum height instantly from initial speed: if the only force is gravity, set (1/2)mv 0 2 = mg h max to get h max = v 0 2/(2g). Likewise, the maximum spring extension under a mass dropped onto a spring follows from equating kinetic and spring potential at the instantary stop. Maximum compression x max of the spring. Application: Maximum spring compression by a falling mass (no bounce, no losses) Take U = 0 at top of spring (before contact). Energy at release: U = mgh 0. At max compression: U = −mg x max (lower position) + (1/2)k x max 2, K = 0. Mass m = 0.40 kg Drop height h 0 = 0.50 m above uncompressed spring Spring constant k = 800 N/m g = 9.8 m/s 2 Initial potential energy. Gravitational plus spring potential at lowest point. Quadratic in x max. Positive root is physical. Drop on spring energy balance medium When mixing mgh with −GMm/r, ensure you use them in their valid ranges. For heights h ≪ R E , mgh is fine. For satellite altitudes, switch to −GMm/r to get accurate energy differences. A good rule: if g changes by more than a few percent over the height range, use the general formula. tip Edge cases: As r → ∞, U → 0 for gravitational potential with the usual reference. As r → 0, the point-mass formula predicts U → −∞ (a signal that real bodies are not point masses). Near Earth’s surface, as h → 0, Δ U g → 0 smoothly. A final strategic note: Energy methods ignore time unless power or timing is asked. If you need time, combine with average or instantaneous power, P = F·v, or revert to kinematics after finding speeds from energy. This hybrid strategy solves many NEET numericals cleanly. Energy bookkeeping always starts and ends at specific states. Draw a quick picture, mark heights and extensions, and write E 1 + W nc = E 2 beneath it. Insert expressions for U and W nc and solve. This habit keeps track of signs and avoids hidden assumptions. Work–energy with potentials (compact) All conservative effects captured by ΔU; only non-conservative work changes mechanical energy. 10 Use this law when analyzing systems where only conservative forces, such as gravity, are doing work. In fields beyond mechanics, the same logic reappears: electrostatic potential energy, chemical potential, and even gravitational binding energy in astrophysics all rely on defining a convenient reference and tracking energy transfer. Mastering the simple mechanical case builds intuition for these advanced topics. Use U s = (1/2)kx 2 for both compression and extension; x is the change from natural length, not total length. Average power equals total work over total time; units are watts (J/s), not joules. Dimension check keeps mistakes in control. For U s = (1/2)kx 2, units are (N/m)· m 2 = N·m = J. For mgh, units are kg·(m/ s 2 )·m = kg· m 2 / s 2 = J. For average power W/Δt, units are J/s = W. Quick checks like these catch algebra slips before they spread. Finally, practice translating words to energy terms: “smooth/rough,” “rise/fall,” “stretch/compress,” “steady speed.” Smooth means W nc = 0; rough means W nc < 0; rise means +mgh; fall means −mgh; stretch/compress means +(1/2)kx 2; steady speed means ΔK = 0 so power balances weight or drag. Mechanical Energy Sum K + U for a system. Conservative Force Has a potential; work is path-independent. Non-conservative Force Dissipative; work depends on path; reduces K + U. Gravitational Potential Energy Near Earth U = mgh; general form U(r) = −GMm/r. Elastic Potential Energy U s = (1/2)kx 2 for an ideal spring. Reference Level Chosen zero of potential energy. Turning Point Where K = 0 and E = U. Average Power W/Δt, the average rate of doing work. Instantaneous Power P = F·v at a moment. Quick Glossary