Section 3.2: Potential Energy
Potential energy is the energy stored in an object due to its position or configuration. It is the counterpart of kinetic energy and plays a vital role in the conservation of mechanical energy.
Gravitational Potential Energy (PE):
\( PE = m g h \)
Where m is mass, g is the acceleration due to gravity, and h is the height above a chosen reference point.
Elastic Potential Energy (Spring):
\( PE_{spring} = \frac{1}{2} k x^2 \)
Where k is the spring constant and x is the displacement from equilibrium.
Conservative Forces: Forces like gravity and spring forces are conservative because the work done depends only on initial and final positions, not the path taken.
Example 1
A 5 kg box is lifted vertically 2 m. Find the gravitational potential energy gained.
\( PE = m g h = 5 \cdot 9.8 \cdot 2 = 98 \text{ J} \)
Example 2
A spring has a constant of 200 N/m. It is compressed 0.1 m. Find the elastic potential energy stored.
\( PE_{spring} = \frac{1}{2} k x^2 = 0.5 \cdot 200 \cdot 0.1^2 = 1 \text{ J} \)
Practice Problems
- A 10 kg object is lifted 5 m. Find the gravitational potential energy gained.
- A spring with k = 150 N/m is stretched 0.2 m. Find the elastic potential energy.
- An object of mass 3 kg is raised from 2 m to 5 m. Calculate the change in gravitational potential energy.
- A 2 kg book is placed on a shelf 1.5 m high. Find the PE.
- A spring compressed 0.05 m stores 0.25 J. Find the spring constant.
- An object of mass 4 kg falls from 10 m. Calculate the gravitational potential energy lost just before hitting the ground.
- A vertical spring has k = 100 N/m. If 0.3 m compressed, find stored energy.
- An object moves from 3 m to 7 m above ground. Mass = 6 kg. Find ΔPE.
- A spring stores 0.5 J when stretched 0.1 m. Find k.
- A 5 kg object lifted 10 m. Find work done against gravity.