Work, Energy and Power • Topic 2 of 3

Potential Energy & Conservation

Potential energy is stored energy a body possesses because of its position or its configuration. A stretched bow, a raised hammer, and a compressed spring all hold energy that can be released to do work later. Unlike kinetic energy, potential energy is always defined relative to a chosen reference level.

Conservative and non-conservative forces. A force is conservative if the work it does on a body depends only on the starting and ending points, not on the path taken, and the work done in a complete round trip is zero. Gravity and the spring force are conservative. Only for conservative forces can we define a potential energy, through $F=-\frac{dU}{dx}$ (force is the negative slope of the potential energy curve). A force is non-conservative if the work depends on the path; friction and air resistance are the standard examples — they convert mechanical energy into heat, so no potential energy can be assigned to them.

Gravitational potential energy. Near the Earth's surface, lifting a body of mass $m$ through a height $h$ requires work $mgh$ against gravity, which is stored as gravitational PE: $U=mgh$, measured from a chosen zero level. This is why a stone at the top of a cliff can do work on the way down.

Spring (elastic) potential energy. A spring obeying Hooke's law exerts a restoring force $F=-kx$, where $k$ is the spring constant (N/m) and $x$ is the extension or compression from the natural length. The work done in stretching it is the area of the triangle under the $F$-$x$ line, giving $U_{spring}=\frac{1}{2}kx^2$. Notice that $U$ depends on $x^2$, so the spring stores the same energy whether stretched or compressed by the same amount.

  • Restoring force grows linearly: $F=-kx$.
  • Stored energy grows as the square: $U=\frac{1}{2}kx^2$.
  • Doubling the deformation quadruples the stored energy.

Conservation of mechanical energy. When only conservative forces do work, the total mechanical energy (kinetic + potential) of a system stays constant: $KE+PE=\text{constant}$, i.e. $\frac{1}{2}mv^2+U=E$. For a freely falling body, as it descends, PE ($mgh$) steadily converts into KE ($\frac{1}{2}mv^2$), so their sum is unchanged. If a non-conservative force such as friction acts, mechanical energy is not conserved — the lost mechanical energy appears as heat, sound or deformation, but the total energy of the universe is still conserved (law of conservation of energy).

Spring force-extension graph: shaded triangle area equals spring PE = half k x squaredExtension xForce F = kxArea = ½ k x²xkx
Solved Examples
1
Worked Example
A 3 kg block is raised to a height of 8 m above the ground. Find its gravitational potential energy. Take $g=10\ \text{m/s}^2$.
Solution
  1. Step 1: Use $U=mgh$.
  2. Step 2: Substitute $m=3\ \text{kg}$, $g=10\ \text{m/s}^2$, $h=8\ \text{m}$.
  3. Step 3: $U=3\times 10\times 8=240\ \text{J}$.

Answer: $U=240\ \text{J}$

2
Worked Example
A spring of force constant 200 N/m is stretched by 0.1 m. Find the potential energy stored in it.
Solution
  1. Step 1: Use $U=\frac{1}{2}kx^2$.
  2. Step 2: Substitute $k=200\ \text{N/m}$, $x=0.1\ \text{m}$.
  3. Step 3: $U=\frac{1}{2}\times 200\times (0.1)^2=\frac{1}{2}\times 200\times 0.01=1\ \text{J}$.

Answer: $U=1\ \text{J}$

3
Worked Example
A stone of mass 0.5 kg is dropped from a height of 20 m. Using conservation of energy, find its speed just before it hits the ground. Take $g=10\ \text{m/s}^2$.
Solution
  1. Step 1: By conservation of energy, all the PE at the top converts to KE at the bottom: $mgh=\frac{1}{2}mv^2$.
  2. Step 2: Mass cancels: $v^2=2gh=2\times 10\times 20=400$.
  3. Step 3: $v=\sqrt{400}=20\ \text{m/s}$.

Answer: $v=20\ \text{m/s}$

4
Worked Example
If the extension of a spring is doubled, by what factor does the stored elastic potential energy change?
Solution
  1. Step 1: Spring PE is $U=\frac{1}{2}kx^2$, so $U\propto x^2$.
  2. Step 2: Replacing $x$ by $2x$ gives $U'=\frac{1}{2}k(2x)^2=\frac{1}{2}k\cdot 4x^2=4\left(\frac{1}{2}kx^2\right)$.
  3. Step 3: Therefore $U'=4U$.

Answer: The stored energy becomes 4 times.

5
Worked Example
A ball of mass 0.2 kg is thrown vertically upward with a speed of 10 m/s. Using conservation of energy, find the maximum height it reaches. Take $g=10\ \text{m/s}^2$.
Solution
  1. Step 1: At maximum height the ball is momentarily at rest, so all the initial KE becomes PE: $\frac{1}{2}mv^2=mgh$.
  2. Step 2: Mass cancels: $h=\frac{v^2}{2g}=\frac{10^2}{2\times 10}$.
  3. Step 3: $h=\frac{100}{20}=5\ \text{m}$.

Answer: $h=5\ \text{m}$

6
Worked Example
A 2 kg block slides down a frictionless incline from a height of 5 m. Find its kinetic energy and speed at the bottom. Take $g=10\ \text{m/s}^2$.
Solution
  1. Step 1: With no friction, mechanical energy is conserved: PE lost = KE gained, so $KE=mgh$.
  2. Step 2: $KE=2\times 10\times 5=100\ \text{J}$.
  3. Step 3: From $KE=\frac{1}{2}mv^2$, $v=\sqrt{\frac{2\,KE}{m}}=\sqrt{\frac{2\times 100}{2}}=\sqrt{100}=10\ \text{m/s}$.

Answer: $KE=100\ \text{J}$, $v=10\ \text{m/s}$

Key Points

  • A conservative force does path-independent work and zero work over a closed loop; gravity and springs are conservative.
  • Friction and air resistance are non-conservative; they dissipate mechanical energy as heat.
  • Gravitational potential energy near the Earth is $U=mgh$, measured from a chosen reference level.
  • Elastic (spring) potential energy is $U=\frac{1}{2}kx^2$; the restoring force is $F=-kx$.
  • When only conservative forces act, total mechanical energy $KE+PE$ stays constant.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.Which of the following is a non-conservative force?
Explanation: Friction does path-dependent work and dissipates energy as heat, so it is non-conservative.
Q2.The potential energy stored in a spring of constant $k$ stretched by $x$ is:
Explanation: The work to stretch a spring is the triangular area under $F=kx$, giving $U=\frac{1}{2}kx^2$.
Q3.A body falls freely from rest. As it falls, its total mechanical energy:
Explanation: With only gravity (a conservative force) acting, $KE+PE$ is conserved; PE simply converts to KE.
Q4.The relation between a conservative force and its potential energy is:
Explanation: A conservative force is the negative gradient of potential energy: $F=-\frac{dU}{dx}$.
Practice more MCQs → 📝 Assignment · 30 marks