Potential energy

What is potential energy?

In plain words: potential energy is energy stored in a body because of where it is (its position, such as its height) or how it is arranged (its configuration, such as a stretched or compressed spring). When the constraints and forces allow motion, this energy can be converted into kinetic energy or other forms; a system at rest in stable equilibrium does not spontaneously release all its potential energy. Strictly, gravitational potential energy belongs to the Earth–body system; speaking of the body’s energy is shorthand for its energy in the Earth’s field.

NCERT gives two vivid examples. A stretched bow-string stores potential energy; released, it sends the arrow off at great speed. The Earth’s crust has fault lines that behave like compressed springs, storing a large amount of potential energy; an earthquake happens when these fault lines readjust and release it.

Gravitational potential energy near the Earth

Near the Earth’s surface, the gravitational force on a ball of mass m is mg, downward, and g can be treated as constant as long as the height h is very small compared with the Earth’s radius. Lift the ball slowly through a height h: the work done against gravity is mgh, and this work is stored as potential energy.

Gravitational potential energy of an object, as a function of the height h, is denoted by V(h) and it is the negative of work done by the gravitational force in raising the object to that height.

NCERT Physics Part I (Class 11), Section 5.7, page 77

Here h is measured upward from a chosen reference level, where V = 0. (Many books write U for potential energy; NCERT uses V.) Only changes in potential energy matter in a problem, so you may choose the reference level wherever it is convenient: the ground, a table top or the lowest point of the motion. Below the reference level, h is negative and so is V.

When the ball falls freely from rest at height h, neglecting air resistance, it reaches the reference level (here the ground, h = 0) with v² = 2gh. Multiplying by m/2:

The potential energy at the top has become kinetic energy at the bottom.

Force from potential energy. Treating h as a variable, the gravitational force is the negative of the slope of V(h):

The minus sign says the force is downward, towards lower potential energy. In one dimension, for any force that has a potential energy V(x):

Like work and kinetic energy, potential energy has dimensions [M L^² T^⁻²] and is measured in joules.

Conservative and non-conservative forces

Potential energy can be defined only for some forces:

Physically, the notion of potential energy is applicable only to the class of forces where work done against the force gets ‘stored up’ as energy.

NCERT Physics Part I (Class 11), Section 5.7, page 77

These are the conservative forces (see conservation of mechanical energy). Gravity and the spring force are conservative; friction and air resistance are not. Air resistance depends on the velocity, so its work depends on how the body moves.

The work done by a conservative force such as gravity depends on the initial and final positions only.

NCERT Physics Part I (Class 11), Section 5.7, page 78

So for a conservative force:

• The work done between two points is the same for every path between them. A block lifted 3 m straight up, or pushed up a smooth ramp to the same height, has the same work done on it by gravity.
• The work done around any closed path, back to the starting point, is zero.
• The change in potential energy is the negative of the work done by the force. For an infinitesimal displacement:

Over a finite displacement, add up the small pieces:

The constant you get on integrating is just your choice of reference level. This is how a potential energy is found from a given force.

A smooth inclined plane shows the idea of path independence. Released from rest at the top of a frictionless incline of height h, a body reaches the bottom with speed $\sqrt{2gh}$, whatever the angle of the incline: the work done by gravity depends only on the height fallen.

For a non-conservative force such as friction, the work done depends on the path. Drag a box around a room and back to where it started: friction has done negative work all the way, and the total is not zero. That energy is not stored; it is turned into heat. So friction has no potential energy.

Potential energy in daily life

• Water stored in a dam. Water held at a height has gravitational potential energy, which turns into kinetic energy as it falls and then drives turbines in a hydroelectric station.
• A bow, a catapult, a stretched rubber band. Elastic potential energy stored by stretching is released as the kinetic energy of the arrow or stone.
• A wound clock spring. Energy stored in the coiled spring is released slowly to drive the clock.
• A book on a high shelf. It has more potential energy than on a low one, which is why a fall from higher up is more damaging.

Solved examples

Common mistakes to avoid

Quick revision summary

• Potential energy is energy stored because of a body’s position or configuration, such as its height or the stretch of a spring.
• Near the Earth, gravitational potential energy is mgh, measured from a chosen reference level; only changes in it matter.
• In one dimension the force is the negative slope of the potential energy, and points towards lower potential energy.
• A force is conservative if its work depends only on the end points, and is zero around any closed path. Gravity and the spring force are conservative; friction is not.
• For a conservative force, the change in potential energy is the negative of the work done by the force.

Questions students ask

Can potential energy be negative?

Yes, potential energy can be negative, because only its changes have meaning and its value depends on where you put the zero. With V = mgh, a body below the reference level has a negative h and a negative potential energy. Outside a spherically symmetric Earth, gravitational potential energy with zero at infinity is −GMm/r, where r is distance from the Earth’s centre, so it is negative throughout that exterior region.

What is the SI unit and dimensional formula of potential energy?

Like all energy, potential energy is measured in joules (J), and its dimensional formula is [M L^² T^⁻²]. You can check this from V = mgh: [M][L T^⁻²][L] = [M L^² T^⁻²].

How are potential energy and force related?

In one dimension, a conservative force is the negative slope of its potential energy, F = −dV/dx, and the other way round the change in potential energy is the negative of the force’s work, $\Delta V = -\int F\,dx$. For gravity near the Earth, V = mgh gives F = −mg: a downward force of size mg. Where dV/dx = 0 the force is zero: a minimum of V is a point of stable equilibrium and a maximum a point of unstable equilibrium.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. A 50 kg student climbs to a roof 10 m above the ground. Find the gain in her gravitational potential energy. Take g = 9.8 m s⁻².

   Show answerHide answer

   $$\begin{aligned} \Delta V &= mgh \\ &= 50 \times 9.8 \times 10 \\ &= 4900\ \text{J} \end{aligned}$$

2. Q2. The potential energy of a particle is V(x) = 5x³ J, with x in metres. Find the force on it at x = 2 m.

   Show answerHide answer

   F = −dV/dx = −15x². At x = 2 m, F = −15 × 4, that is −60 N: 60 N towards negative x.

3. Q3. Why can we define a potential energy for gravity but not for friction?

   Show answerHide answer

   The work done by gravity depends only on the initial and final heights, and is zero around a closed path, so the work done against it is stored and can be got back: it is conservative. The work done by friction depends on the path, is not zero around a closed path, and is lost as heat rather than stored, so friction is non-conservative and has no potential energy.