Learn what potential energy is and how to calculate the potential energy of gravity and also that of a spring.

### Key Equations

**Gravitational potential energy**

\begin{align*}U_g= m g h \begin{cases} h & \text{height above the ground in meters(m)}\\ g & \text{acceleration due to gravity,}\ 9.8\ \text{m/s}^{2}\\ U_{g} & \text{Potential energy of gravity (in Joules; 1 J} = 1 \text{kg} \ \text{m}^{2}/\text{s}^{2}\text{)} \end{cases}\end{align*}

**Spring potential energy**

\begin{align*} U_{sp} = \tfrac{1}{2}k{\Delta x}^2 \begin{cases} k & \text{spring constant measured in Newtons(N) per meters(m)}\\ x & \text{amount spring is displaced from resting position}\\ U_{sp} & \text{potential energy of a spring (in Joules; 1 J} = 1 \text{kg} \ \text{m}^{2}/\text{s}^{2}\text{)} \end{cases} \end{align*}

### Guidance

#### Example 1

A 2 kg block of wood is suspended 5m above a spring of spring constant 3000 N/m. When the block is dropped on the spring, how far will the spring be compressed from it's equilibrium position.

##### Solution

We can solve for the distance the spring will be compressed using conservation of energy. In this problem, the gravitational potential energy of the block will be turned into spring potential energy.

\begin{align*} U_g&=U_{sp} && \text{start with by setting the gravitational potential energy equal to the spring potential energy}\\ mgh&=\frac{1}{2}k\Delta x^2 && \text{substitute the equations for gravitational and spring potential energy}\\ \Delta x&=\sqrt{\frac{2mgh}{k}} && \text{solve for } \Delta x\\ \Delta x&=\sqrt{\frac{2*2\;\text{kg} * 9.8\;\text{m/s}^2 * 5\;\text{m}}{3000\;\text{N/m}}} && \text{plug in the known values}\\ \Delta x&=.26\;\text{m}\\ \end{align*}

### Explore More

- If you lift a 30 kg weight 0.5 meters, how much Potential energy has it gained?
- A spring has a spring constant
*k*equal to 200 N/m. If it is stretched 0.4 m, what is its potential energy? - A 12 kg box is resting on a table 1.5 m off the ground. It is then lifted up to 3.2 m off the ground. What is its increase in potential energy?

#### Answers

(using g = 10 m/s^{2})

- 150 J
- 16 J
- 204 J