# 6.5: Potential Energy

- Page ID
- 2862

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## At what point does this diver have the most energy?

This diver is going to jump from the end of the diving board. After she dives down and is falling toward the water, she’ll have kinetic energy, or the energy of moving matter. But even as she is momentarily stopped high above the water, she has energy. Do you know why?** **

## Stored Energy

The diver has energy because of her position high above the pool. The type of energy she has is called potential energy. **Potential energy** is energy that is stored in a person or object. Often, the person or object has potential energy because of its position or shape.

**Q:** What is it about the diver’s position that gives her potential energy?

**A:** Because the diver is high above the water, she has the potential to fall toward Earth because of gravity. This gives her potential energy.

## Gravitational Potential Energy

Potential energy due to the position of an object above Earth’s surface is called *gravitational potential energy.* Like the diver on the diving board, anything that is raised up above Earth’s surface has the potential to fall because of gravity. You can see another example of people with gravitational potential energy in the **Figure** above.

## Elastic Potential Energy

Potential energy due to an object’s shape is called *elastic potential energy*. This energy results when an elastic object is stretched or compressed. The farther the object is stretched or compressed, the greater its potential energy is. A point will be reached when the object can’t be stretched or compressed any more. Then it will forcefully return to its original shape.

Shooting an arrow from a bow, as shown in the image above, requires work done on the bow by the shooter's arm to bend the bow and thus produce potential energy. The release of the bow converts the potential energy of the bent bow into the kinetic energy of the flying arrow.

Play around with the simulation below to adjust the stretch distance and elastic constant of a bow to see its effect on the motion of the arrow:

Interactive Element

## Other Forms of Potential Energy

If you hold two positive charges near each other, their *electromagnetic*** potential energy** pushes them apart when you let go. Potential energy is stored in

**chemical bonds**(

**). When these bonds are broken, the excess**

*chemical potential energy***energy**is seen as molecular motion and

**heat**.

## Calculating Potential Energy

If a cannon ball is fired straight up into the air, it begins with a high kinetic energy. As the cannon ball rises, it slows down due to the force of gravity pulling it toward the earth. As the ball rises, its gravitational potential energy is increasing and its kinetic energy is decreasing. When the cannon ball reaches the top of its arc, its kinetic energy is zero and its potential energy is at the maximum. As gravity continues to pull the cannon ball toward the earth, the ball will fall downwards, causing its height to decrease and its speed to increase. The ball's potential energy decreases and its kinetic energy increases. When the ball returns to its original height, its kinetic energy will be the same as when it started upward.

When work is done on an object, the work may be converted into either kinetic or potential energy. Work resulting in motion is caused when the work is converted into kinetic energy, while work resulting in a change of position is caused by a conversion into potential energy. Work is also spent overcoming friction and that work would be converted into heat, but we will consider primarily frictionless systems.

If we consider the potential energy of a bent stick or a stretched rubber band, the potential energy can be calculated by multiplying the force exerted by the stick or rubber band by the distance over which the force will be exerted. The formula for calculating this potential energy looks exactly like the formula for calculating work done: W=Fd. The only difference is that work is calculated when the object actually moves and potential energy is calculated when the system is still at rest, before any motion actually occurs.

In the case of gravitational potential energy, the force exerted by the object is its weight and the distance it can travel is its height above the earth. Since the weight of an object is calculated by W=mg, then gravitational potential energy can be calculated by PE=mgh, where *m* is the mass of the object, g is the acceleration due to gravity, and *h* is the height the object will fall.

## Examples

Example 6.5.1

A 3.00 kg object is lifted from the floor and placed on a shelf that is 2.50 m above the floor.

(a) What was the work done in lifting the object?

(b) What is the gravitational potential energy of the object sitting on the shelf?

**Solution**

weight of the object=mg=(3.00 kg)(9.80 m/s2)=29.4 N

(a) W=Fd=(29.4 N)(2.50 m)=73.5 J

(b) PE=mgh=(3.00 kg)(9.80 m/s2)(2.50 m)=73.5 J

Example 6.5.2

A pendulum is constructed from a 7.58 kg bowling ball hanging on a 3.00 m long rope. The ball is pulled back until the rope makes an angle of 45∘ with the vertical.

(a) What is the potential energy of the ball?

**Solution**

You can use trigonometry to find the vertical height of the ball in the pulled back position. This vertical height is found to be 0.877 m.

(a) PE=mgh=(7.58 kg)(9.80 m/s2)(0.877 m)=65.1 J

A trampoline can store elastic potential energy just like a rubber band, spring or bow. The amount of energy stored relies on how far it is stretched. This energy is transferred into kinetic energy of the jumper as the are launched into the air. The energy is also converted into gravitational potential energy as the jumper gets higher and higher. Jump around in the simulation below to learn more:

Interactive Element

## Summary

- Stored energy is called potential energy.
- Energy may be stored by holding an object elevated in a gravitational field or by holding it while a force is attempting to move it.
- Potential energy may be converted to kinetic energy.
- The formula for gravitational potential energy is PE=mgh.
- In the absence of friction or bending, work done on an object must become either potential energy or kinetic energy or both.

## Review

- A 90.0 kg man climbs hand over hand up a rope to a height of 9.47 m. How much potential energy does he have at the top?
- A 50.0 kg shell was fired from a cannon at earth’s surface to a maximum height of 400. m.
- What is the potential energy at maximum height?
- It then fell to a height of 100. m. What was the loss of PE as it fell?

- A person weighing 645 N climbs up a ladder to a height of 4.55 m.
- What work does the person do?
- What is the increase in gravitational potential energy?
- Where does the energy come from to cause this increase in PE?

## Explore More

Use this resource to answer the questions that follow.

- What is the definition of energy?
- Name two types of potential energy.
- How is energy transferred from one object to another?

## Additional Resources

Interactive: Roller Coaster

PLIX: Play, Learn, Interact, eXplore: Physics of Archery

Real World Application: Trebuchet

Video:

Study Guide: Energy Study Guide