Login Free Test Start Mock

Physics · Work, Energy and Power

Potential Energy & Conservation of Mechanical Energy

Potential energy is the energy a body stores by virtue of its position or configuration — the stretched bow-string, the raised stone, the compressed spring. NCERT §5.7 makes that idea precise: gravitational potential energy is \(V(h) = mgh\), and the force that stores it satisfies \(F = -\,dV/dx\). §5.8 then shows that whenever only such conservative forces act, the sum of kinetic and potential energy stays constant. This deep-dive develops the conservative-force condition, derives the conservation law, reads it off energy bar-charts for free fall and the pendulum, and works through the free-fall, pendulum and vertical-circle problems NEET keeps returning to.

Potential energy: stored energy of configuration

The word potential suggests a possibility or capacity for action, and potential energy is exactly that — energy that has been stored and is waiting to be released. NCERT opens §5.7 with familiar images: a stretched bow-string holds potential energy, and when released, the arrow flies off at great speed; the fault lines in the Earth's crust behave like compressed springs holding enormous potential energy, released as an earthquake when they readjust.

The unifying statement is precise: potential energy is the stored energy by virtue of the position or configuration of a body. Left to itself, the body releases this stored energy in the form of kinetic energy. Raise a ball and you have done work against gravity; that work is now bankable as motion the instant you let go.

Crucially, the notion applies only to a special class of forces — those for which the work done against the force gets stored up and can later be fully recovered as kinetic energy. For such forces we can define a potential energy function; for others, such as friction, we cannot. That distinction drives the whole of this topic.

Gravitational potential energy = mgh

Take a ball of mass \(m\) near the Earth's surface, where \(g\) may be treated as constant — "near" meaning the height \(h\) is very small compared with the Earth's radius \(R_E\) (so \(h \ll R_E\) and the variation of \(g\) is ignored). The gravitational force on the ball is \(mg\), directed downward. Raise the ball through a height \(h\). The work done by the external agency against the gravitational force is \(mgh\), and this work is stored as potential energy.

NCERT defines the gravitational potential energy \(V(h)\) as the negative of the work done by the gravitational force in raising the object to that height:

\[ V(h) = mgh \]

The NIOS lesson reaches the same expression by lifting a mass from height \(h_1\) to \(h_2\): the work done against gravity is \(W = mgh\) with \(h = h_2 - h_1\), stored as gravitational potential energy. Both texts stress that the choice of the zero level is arbitrary; normally the Earth's surface is taken as the reference point of zero potential energy, but only the change in height matters physically.

Ground · V = 0 h m mg Work against gravity W = mgh stored as V(h) = mgh
Raising a mass m through height h stores work mgh as gravitational potential energy. Released, it returns as kinetic energy. (NCERT §5.7)

What is stored can be recovered. Release the ball and it falls with increasing speed; just before it hits the ground its speed obeys the kinematic relation \(v^2 = 2gh\). Multiplying by \(m/2\),

\[ \tfrac{1}{2}mv^2 = mgh \]

so the gravitational potential energy at height \(h\) reappears exactly as kinetic energy at the ground. Potential and kinetic energy share the same dimensions, \([\mathrm{ML^2T^{-2}}]\), and the same SI unit, the joule. This single line — PE at the top equalling KE at the bottom — is the seed of the conservation principle developed below.

The conservative-force condition: F = −dV/dx

If \(h\) is treated as a variable, the gravitational force equals the negative of the derivative of \(V(h)\):

\[ F = -\frac{d}{dh}V(h) = -\frac{d}{dh}(mgh) = -mg \]

The negative sign signals that the gravitational force points downward, opposite to the direction of increasing height. NCERT generalises this to any one-dimensional force: a potential energy \(V(x)\) is defined whenever the force can be written as

\[ F(x) = -\frac{dV(x)}{dx} \qquad\Longleftrightarrow\qquad \Delta V = -F(x)\,\Delta x \]

Read physically: the force always points "downhill" on the potential energy curve, toward lower \(V\). The change in potential energy is the negative of the work done by the force. This is the analytic test for whether a force has an associated potential energy at all.

Conservative vs non-conservative forces

A force is conservative if the work it does on an object is path-independent and depends only on the end points; equivalently, if the work it does around any closed path is zero; equivalently again, if it can be derived from a potential energy by \(F = -dV/dx\). NCERT gives all three as equivalent definitions. The work done by such a force between two points is simply \(W = V(x_i) - V(x_f)\) — the end points are all that matter. As a clean illustration, a body released from rest at the top of a smooth incline of height \(h\) reaches the bottom with speed \(\sqrt{2gh}\) and kinetic energy \(mgh\) irrespective of the angle of inclination; the path does not enter.

A non-conservative force fails these tests. Friction is the standard example: drag a block from A to B along a long path and it loses more energy than along a short one, so the work depends on the path. NCERT states it flatly — work done by friction over a closed path is not zero, and no potential energy can be associated with friction. The energy it removes is not stored; it is dissipated as heat and sound.

Conservative forces

  • Work depends only on the end points, not on the path taken.
  • Work done over any closed loop is zero.
  • Derivable from a potential energy: \(F = -dV/dx\).
  • Energy is stored and fully recoverable as kinetic energy.
  • Examples (NIOS): gravitational force, elastic (spring) force, electrostatic force.

Non-conservative forces

  • Work depends on the path between the two points.
  • Work over a closed loop is not zero.
  • No potential energy function can be defined.
  • Energy is dissipated — lost from the mechanical account as heat or sound.
  • Example: force of friction; air resistance behaves the same way.

Conservation of mechanical energy

Now the central result. Suppose a body undergoes a small displacement \(\Delta x\) under a conservative force \(F\). The work–energy theorem gives the change in kinetic energy,

\[ \Delta K = F(x)\,\Delta x \]

But for a conservative force we may define \(V(x)\) such that \(-\Delta V = F(x)\,\Delta x\). Substituting,

\[ \Delta K = -\Delta V \quad\Longrightarrow\quad \Delta K + \Delta V = 0 \quad\Longrightarrow\quad \Delta(K + V) = 0 \]

The quantity \(K + V\) does not change. Integrated over the whole path from \(x_i\) to \(x_f\), this reads

\[ K_i + V(x_i) = K_f + V(x_f) \]

The sum \(E = K + V(x)\) is the total mechanical energy. Individually \(K\) and \(V\) may swing from point to point, but their sum is fixed. NCERT states the principle in one sentence:

The total mechanical energy of a system is conserved if the forces doing work on it are conservative.

The aptness of the name "conservative force" is now plain — these are precisely the forces that conserve mechanical energy. The conservation principle is not a new law but a consequence of the work–energy theorem applied to conservative forces.

Energy bar-charts: free fall and the pendulum

The conservation law is easiest to feel as a moving bar-chart in which the total height of the stack never changes. NCERT's Fig. 5.5 considers a ball of mass \(m\) dropped from a cliff of height \(H\). Its mechanical energies at heights \(H\), \(h\) and the ground are

\[ E_H = mgH,\qquad E_h = mgh + \tfrac{1}{2}mv_h^2,\qquad E_0 = \tfrac{1}{2}mv_f^2 \]

Since gravity is conservative, \(E_H = E_h = E_0\). At the top the energy is purely potential; partway down it is part potential, part kinetic; at the ground it is fully kinetic. Setting \(E_H = E_0\) gives \(mgH = \tfrac{1}{2}mv_f^2\), i.e. \(v_f = \sqrt{2gH}\) — the freely-falling-body result, with the mass cancelling out.

Total mechanical energy E = mgH (constant) PE At top H PE KE At height h KE At ground
Free fall from height H: the bar's total height (total mechanical energy) is fixed; only the PE–KE split shifts. PE drains into KE as the ball descends. (NCERT Fig. 5.5)

The frictionless pendulum tells the same story sideways. At the extreme swing the bob is momentarily at rest — kinetic energy zero, potential energy maximum. At the lowest point the bob is fastest — kinetic energy maximum, potential energy at its minimum. Everywhere between, \(K + V\) is the same constant. NIOS verifies the law directly for free fall by computing \(K + V = mgh_1 + mgh_2 = mgh\) at an intermediate point, equal to the energy at the top.

PE max KE = 0 PE max KE = 0 KE max PE min Lowest point: fastest
A frictionless pendulum: potential energy peaks at the extremes (bob at rest), kinetic energy peaks at the bottom (bob fastest). The sum K + V is the same everywhere along the swing.

What friction does to the energy book

The conservation law has a precondition: only conservative forces do work. The moment a non-conservative force such as friction or air resistance enters, mechanical energy is no longer conserved. NCERT modifies the bookkeeping cleanly. If the forces are a conservative \(F_c\) and a non-conservative \(F_{nc}\), the work–energy theorem gives \((F_c + F_{nc})\Delta x = \Delta K\), and since \(F_c \Delta x = -\Delta V\),

\[ \Delta(K + V) = F_{nc}\,\Delta x \quad\Longrightarrow\quad E_f - E_i = W_{nc} \]

where \(W_{nc}\) is the work done by the non-conservative forces along the path. Unlike conservative work, \(W_{nc}\) depends on the route taken. For friction \(W_{nc}\) is negative, so the mechanical energy steadily decreases — the missing energy reappears as thermal energy. NIOS makes the picture concrete: a block given speed \(v\) on a rough surface stops at B with neither kinetic nor potential energy; its energy has changed form, dissipated as heat by friction's negative work.

i
Related drill

The spring force is the other classic conservative force, with \(V(x) = \tfrac{1}{2}kx^2\) — see potential energy of a spring for the full treatment.

Worked example 1 — free-fall energy split

Conservation · Free fall

A ball of mass \(m\) is dropped from rest at a cliff of height \(H\) (gravity only, no air resistance). Find (a) its speed at the ground, and (b) the fraction of the total energy that is kinetic when the ball has fallen to a point where the remaining height above the ground is \(h\).

(a) Mechanical energy is conserved because only gravity acts. Take \(V = 0\) at the ground. At the top \(E_H = mgH\) (all potential); at the ground \(E_0 = \tfrac{1}{2}mv_f^2\) (all kinetic). Conservation gives \(mgH = \tfrac{1}{2}mv_f^2\), so \(v_f = \sqrt{2gH}\). The mass cancels, exactly as NCERT obtains for a freely falling body.

(b) At remaining height \(h\) the potential energy is \(V = mgh\), so the kinetic energy is \(K = E - V = mgH - mgh = mg(H - h)\). The kinetic fraction is therefore \(\dfrac{K}{E} = \dfrac{mg(H-h)}{mgH} = 1 - \dfrac{h}{H}\). At \(h = H\) the fraction is 0 (all potential); at \(h = 0\) it is 1 (all kinetic) — the bar-chart in words.

Worked example 2 — pendulum bob speed at the bottom

NCERT Exercise 5.18 (adapted)

The bob of a pendulum is released from the horizontal position. The length of the pendulum is \(L = 1.5~\text{m}\). It dissipates 5% of its initial energy against air resistance. With what speed does the bob arrive at the lowermost point? (Take \(g = 10~\text{m s}^{-2}\).)

Set the zero of PE at the lowest point. Released from the horizontal, the bob starts a height \(L\) above the lowest point, so its initial energy is purely potential: \(E_i = mgL\).

Account for the non-conservative loss. Air resistance is non-conservative, so mechanical energy is not conserved. The bob keeps \(95\%\) of the initial energy as kinetic energy at the bottom: \(\tfrac{1}{2}mv^2 = 0.95\,mgL\).

Solve. \(v = \sqrt{2(0.95)gL} = \sqrt{2(0.95)(10)(1.5)} = \sqrt{28.5} \approx 5.34~\text{m s}^{-1}\). Had there been no air resistance, the bob would arrive at \(\sqrt{2gL} = \sqrt{30} \approx 5.48~\text{m s}^{-1}\) — the 5% dissipation lowers the arrival speed only slightly because energy scales as \(v^2\).

Worked example 3 — completing a vertical circle

NCERT Example 5.7

A bob of mass \(m\) hangs on a light string of length \(L\). It is given a horizontal speed \(v_0\) at the lowest point A such that it just completes a vertical circle, the string becoming slack only at the topmost point C. Find (i) \(v_0\), and (ii) the speed \(v_C\) at the top.

Forces and energy. Two forces act: gravity and the string tension. The tension does no work (always perpendicular to the motion), so the potential energy is gravitational only and the total mechanical energy is conserved. Take \(V = 0\) at the lowest point A.

Condition at the top C. "Just completes" means the string slackens exactly at C, so the tension there is zero and gravity alone supplies the centripetal force: \(mg = \dfrac{mv_C^2}{L}\), giving \(v_C^2 = gL\).

Conservation between A and C (C is a height \(2L\) above A): \(\tfrac{1}{2}mv_0^2 = \tfrac{1}{2}mv_C^2 + mg(2L)\). Substituting \(v_C^2 = gL\): \(\tfrac{1}{2}v_0^2 = \tfrac{1}{2}gL + 2gL = \tfrac{5}{2}gL\), so \(v_0 = \sqrt{5gL}\).

Answer: \(v_0 = \sqrt{5gL}\) at the bottom and \(v_C = \sqrt{gL}\) at the top — the standard minimum-speed result for a vertical circle on a string.

Quick recap

Potential energy & conservation in one breath

  • Potential energy is stored energy of position or configuration; gravitational PE near Earth is \(V(h) = mgh\).
  • A force is conservative if its work is path-independent (zero over a closed loop) and obeys \(F = -dV/dx\).
  • Gravity, spring and electrostatic forces are conservative; friction and air resistance are not.
  • If only conservative forces do work, \(K + V\) = constant: \(K_i + V(x_i) = K_f + V(x_f)\).
  • Free fall: \(v = \sqrt{2gH}\) (mass cancels). Vertical circle on a string: \(v_0 = \sqrt{5gL}\), \(v_{\text{top}} = \sqrt{gL}\).
  • With friction/air drag, mechanical energy is not conserved: \(E_f - E_i = W_{nc}\), and \(W_{nc}\) depends on the path.
  • The zero of potential energy is arbitrary — fix it once per problem and keep it.

NEET PYQ Snapshot — Potential Energy & Conservation

Three PYQs that turn on conservation of mechanical energy and the vertical-circle condition. Same routine: fix the PE zero, equate energies, add the centripetal condition where the circle must close.

NEET 2021

A particle is released from height \(S\) from the surface of the Earth. At a certain height its kinetic energy is three times its potential energy. The height from the surface and the speed of the particle at that instant are respectively:

  1. \(\dfrac{S}{4},\ \sqrt{\dfrac{3gS}{2}}\)
  2. \(\dfrac{S}{4},\ \sqrt{\dfrac{3gS}{2}}\)
  3. \(\dfrac{S}{2},\ \sqrt{\dfrac{3gS}{2}}\)
  4. \(\dfrac{S}{4},\ \sqrt{\dfrac{gS}{2}}\)
Answer: (1) S/4, √(3gS/2)

Conservation-driven. At height \(y\), \(U = mgy\) and \(K = 3U = 3mgy\). Total energy \(= mgS\) (released from rest at height \(S\)), so \(mgy + 3mgy = mgS \Rightarrow y = S/4\). Speed from \(K = \tfrac{1}{2}mv^2 = 3mg(S/4)\): \(v = \sqrt{3gS/2}\).

NEET 2018

A body initially at rest slides along a frictionless track from a height \(h\) and just completes a vertical circle of diameter \(AB = D\). The height \(h\) equals:

  1. \(\dfrac{3}{2}D\)
  2. \(\dfrac{7}{5}D\)
  3. \(D\)
  4. \(\dfrac{5}{4}D\)
Answer: (4) 5D/4

Vertical-circle condition + conservation. At the top of the loop, minimum speed satisfies \(v^2 = gr = g(D/2)\). The track is frictionless, so energy is conserved between the start (height \(h\)) and the top of the loop (height \(D\)): \(mg(h - D) = \tfrac{1}{2}mv^2 = \tfrac{1}{2}m\,g(D/2)\). Solving, \(h - D = D/4\), so \(h = 5D/4\).

NEET 2016

What is the minimum velocity with which a body of mass \(m\) must enter a vertical loop of radius \(R\) so that it can complete the loop?

  1. \(\sqrt{2gR}\)
  2. \(\sqrt{3gR}\)
  3. \(\sqrt{5gR}\)
  4. \(\sqrt{gR}\)
Answer: (3) √(5gR)

Same machinery as Example 5.7. At the top the minimum speed gives \(v_{\text{top}}^2 = gR\). Conservation from bottom to top (height \(2R\)): \(\tfrac{1}{2}mv^2 = \tfrac{1}{2}mv_{\text{top}}^2 + mg(2R)\). With \(v_{\text{top}}^2 = gR\), this yields \(v = \sqrt{5gR}\) — the minimum entry speed at the lowest point of the loop.

FAQs — Potential Energy & Conservation

Short answers to the conservation questions NEET aspirants get wrong most often.

Why is the gravitational potential energy taken as the negative of the work done by gravity?
Potential energy is the energy stored when you do work against a conservative force. NCERT defines V(h) as the negative of the work done by the gravitational force in raising the object to height h. When you raise a ball, the work done by you against gravity is +mgh, and this is exactly what gets stored. Equivalently, the work done by gravity itself is −mgh, so V(h) = −(−mgh) = mgh. The sign convention keeps the bookkeeping consistent: a positive PE corresponds to energy you can recover as kinetic energy on release.
Is the zero of potential energy fixed by physics?
No. NCERT states that the zero of the potential energy is arbitrary and is set by convenience. For the constant gravitational force mg we usually take V = 0 at the Earth's surface; for the spring we take V = 0 at the natural length. Only the change in potential energy enters the physics. Once you fix the zero in a given problem you must stick to it throughout — in NCERT's words, "you cannot change horses in midstream."
What exactly is a conservative force?
A force is conservative if the work it does on an object is path-independent and depends only on the end points, or equivalently if the work it does over any closed path is zero. A third equivalent test is that it can be derived from a potential energy function as F(x) = −dV/dx. Gravity, the spring force and the electrostatic force are conservative. Friction is not — work done by friction over a closed path is not zero, and no potential energy can be associated with it.
When is mechanical energy conserved?
The total mechanical energy of a system is conserved if the only forces doing work on it are conservative. For a freely falling ball or a frictionless pendulum, KE + PE stays constant — kinetic energy converts to potential and back, but their sum does not change. The instant a non-conservative force such as friction or air resistance does work, mechanical energy is no longer conserved; the lost mechanical energy reappears as heat or sound.
Does the speed of a freely falling body depend on its mass?
No. Equating mgH to ½mv² gives v = √(2gH); the mass cancels. NCERT obtains exactly this result from conservation of mechanical energy for a ball dropped from a cliff of height H. The same cancellation explains why a heavy and a light object, released together in the absence of air resistance, reach the ground at the same speed.
What is the minimum speed needed at the lowest point to complete a vertical circle?
For a bob on a light string (or a body on the inside of a vertical loop), the string becomes slack at the top when the tension there falls to zero, so gravity alone supplies the centripetal force: mg = mv²/L at the top, giving v_top² = gL. Applying conservation of mechanical energy between the bottom and the top (height 2L apart) gives the minimum launch speed at the lowest point as v₀ = √(5gL), the standard NEET result.
Where is the kinetic energy maximum and where is the potential energy maximum for a swinging pendulum?
For a frictionless pendulum the potential energy is maximum at the extreme positions (where the bob is momentarily at rest, so kinetic energy is zero) and the kinetic energy is maximum at the lowest point (where the bob is fastest, so potential energy is at its minimum). At every point in between, KE + PE equals the same constant total — the energy continuously trades between the two forms.
Related Topics
Work, Energy and Power — Parent chapter Full chapter map: work, kinetic energy, potential energy, power, collisions. Work — Definition & Calculation W = F·d cos θ; positive, negative and zero work. Kinetic Energy & Work–Energy Theorem K = ½mv²; K_f − K_i = W_net; the theorem behind conservation. Work Done by a Variable Force Work as the area under F(x); the integral form. Potential Energy of a Spring V(x) = ½kx²; the other classic conservative force. Power Time rate of work; P = F·v; watt and horsepower. Elastic & Inelastic Collisions Momentum vs kinetic-energy conservation in collisions. Circular Motion & Centripetal Force The centripetal condition used in vertical-circle problems.