S3.6 Potential energy: examples
- In this course, you will meet a few different types of potential
energy:
- energy stored in a spring;
- gravitational potential energy;
- energy stored in a wave;
- There are many, many other sorts of
potential energy:
- energy stored in chemical bonds;
- electrical energy stored in a capacitor;
- magnetic energy stored in a field;
- energy stored in excited states of atoms;
[A] Linear (spring) force: potential energy
- The force a spring exerts on a body is proportional to the displacement of the spring (S2.9).
Note: convention dictates we choose the reference point to be the displacement from the equilibrium length. |

Worked example
The scale of a certain spring balance reads from zero to 1200N and is 0.1m long.
What is the potential energy of the spring when :
- [(a)] it is stretched 0.1m? 0.05m?
- [(b)] a 60kg mass hangs from the spring.
Solution
A force of -1200N fully stretches the spring to 0.1m. Thus, using Hooke’s law, the spring constant is k=-F/x=1200/0.1=1.2×104N/m.
- [(a)] The potential energy for any extension is given by U(x)=kx2/2. Thus one finds U(0.1)=60J, U(0.05)=15J.
- [(b)] For a 60kg mass the extension will be x=mg/k=0.05m and hence the potential energy is again U(0.05)=15J.
[B] Gravitational potential energy of a body near the earth’s surface
- Near the earth’s surface the gravitational force is constant
(S2.5).
Key Point 3.13
Gravitational potential energy. The potential energy of a body at height h above some reference point (say the earth’s surface) is
U=mgh
[C] Gravitational potential energy of two point masses
- The gravitational force acting between two masses is proportional
to each mass, and inversely proportional to the square of their
separation (Key Point 2.4)
Key Point 3.14
Gravitational Potential Energy. The gravitational potential energy of a particle of mass m2 a distance r from a body of mass m1 is
where G is the gravitational constant: 6.673 × 10-11Nm2kg-2

Analysis
calculating the associated force as before, we find
![\[ F = \frac{Gm_1m_2}{r^2} \;, \]](mastermathpng-4.png)
the attractive gravitational force between two bodies. (S2.10)
Note: By definition, two bodies infinitely far apart have a potential energy of zero: this is the reference point, infinite separation.
Learning Resources
![]() | HRW Chapter 8.4 |
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