# Learning Objectives

### Learning Objectives

By the end of this section, you will be able to do the following:

- Explain the law of conservation of energy in terms of kinetic and potential energy
- Perform calculations related to kinetic and potential energy. Apply the law of conservation of energy

law of conservation of energy |

# Mechanical Energy and Conservation of Energy

### Mechanical Energy and Conservation of Energy

We saw earlier that mechanical energy can be either potential or kinetic. In this section we will see how energy is transformed from one of these forms to the other. We will also see that, in a closed system, the sum of these forms of energy remains constant.

Quite a bit of potential energy is gained by a roller coaster car and its passengers when they are raised to the top of the first hill. Remember that the *potential* part of the term means that energy has been stored and can be used at another time. You will see that this stored energy can either be used to do work or can be transformed into kinetic energy. For example, when an object that has gravitational potential energy falls, its energy is converted to kinetic energy. Remember that both work and energy are expressed in joules.

Refer back to Figure 9.3. The amount of work required to raise the TV from point A to point B is equal to the amount of gravitational potential energy the TV gains from its height above the ground. This is generally true for any object raised above the ground. If all the work done on an object is used to raise the object above the ground, the amount work equals the object’s gain in gravitational potential energy. However, note that because of the work done by friction, these energy–work transformations are never perfect. Friction causes the loss of some useful energy. In the discussions to follow, we will use the approximation that transformations are frictionless.

Now, let’s look at the roller coaster in Figure 9.6. Work was done on the roller coaster to get it to the top of the first rise; at this point, the roller coaster has gravitational potential energy. It is moving slowly, so it also has a small amount of kinetic energy. As the car descends the first slope, its *PE* is converted to *KE*. At the low point much of the original *PE* has been transformed to *KE*, and speed is at a maximum. As the car moves up the next slope, some of the *KE* is transformed back into *PE* and the car slows down.

### Virtual Physics

#### Energy Skate Park Basics

This simulation shows how kinetic and potential energy are related, in a scenario similar to the roller coaster. Observe the changes in *KE* and *PE* by clicking on the bar graph boxes. Also try the three differently shaped skate parks. Drag the skater to the track to start the animation.

- The mechanical energy of the system increases, provided there is no loss of energy due to friction. The energy would transform to kinetic energy when the speed is increasing.
- The mechanical energy of the system remains constant provided there is no loss of energy due to friction. The energy would transform to kinetic energy when the speed is increasing.
- The mechanical energy of the system increases provided there is no loss of energy due to friction. The energy would transform to potential energy when the speed is increasing.
- The mechanical energy of the system remains constant provided there is no loss of energy due to friction. The energy would transform to potential energy when the speed is increasing.

On an actual roller coaster, there are many ups and downs, and each of these is accompanied by transitions between kinetic and potential energy. Assume that no energy is lost to friction. At any point in the ride, the total mechanical energy is the same, and it is equal to the energy the car had at the top of the first rise. This is a result of the law of conservation of energy, which says that, in a closed system, total energy is conserved—that is, it is constant. Using subscripts 1 and 2 to represent initial and final energy, this law is expressed as

Either side equals the total mechanical energy. The phrase *in a **closed system* means we are assuming no energy is lost to the surroundings due to friction and air resistance. If we are making calculations on dense falling objects, this is a good assumption. For the roller coaster, this assumption introduces some inaccuracy to the calculation.

# Calculations involving Mechanical Energy and Conservation of Energy

### Calculations involving Mechanical Energy and Conservation of Energy

### Tips For Success

When calculating work or energy, use units of meters for distance, newtons for force, kilograms for mass, and seconds for time. This will assure that the result is expressed in joules.

### Watch Physics

#### Conservation of Energy

This video discusses conversion of *PE* to *KE* and conservation of energy. The scenario is very similar to the roller coaster and the skate park. It is also a good explanation of the energy changes studied in the snap lab.

- The speed was the same in the scenario in the animation because the object was sliding on the ice, where there is large amount of friction. In real life, much of the mechanical energy is lost as heat caused by friction.
- The speed was the same in the scenario in the animation because the object was sliding on the ice, where there is small amount of friction. In real life, much of the mechanical energy is lost as heat caused by friction.
- The speed was the same in the scenario in the animation because the object was sliding on the ice, where there is large amount of friction. In real life, no mechanical energy is lost due to conservation of the mechanical energy.
- The speed was the same in the scenario in the animation because the object was sliding on the ice, where there is small amount of friction. In real life, no mechanical energy is lost due to conservation of the mechanical energy.

### Worked Example

#### Applying the Law of Conservation of Energy

A 10 kg rock falls from a 20 m cliff. What is the kinetic and potential energy when the rock has fallen 10 m?

### Strategy

Choose the equation.

List the knowns.

*m* = 10 kg, **v**_{1} = 0, **g** = 9.80

*h*_{1} = 20 m, *h*_{2} = 10 m

Identify the unknowns.

*KE*_{2} and *PE*_{2}

Substitute the known values into the equation and solve for the unknown variables.

Alternatively, conservation of energy equation could be solved for **v**_{2} and *KE*_{2} could be calculated. Note that *m* could also be eliminated.

### Tips For Success

Note that we can solve many problems involving conversion between *KE* and *PE* without knowing the mass of the object in question. This is because kinetic and potential energy are both proportional to the mass of the object. In a situation where *KE* = *PE*, we know that *m***g***h* = (1/2)*m***v**^{2}.

Dividing both sides by *m* and rearranging, we have the relationship

2**g***h* = **v**^{2}.

# Practice Problems

### Practice Problems

A child slides down a playground slide. If the slide is 3 m high and the child weighs 300 N, how much potential energy does the child have at the top of the slide? (Round *g* to $$10\text{}m/{s}^{2}.$$)

- 0 J
- 100 J
- 300 J
- 900 J

A 0.2 kg apple on an apple tree has a potential energy of 10 J. It falls to the ground, converting all of its PE to kinetic energy. What is the velocity of the apple just before it hits the ground?

- 0 m/s
- 2 m/s
- 10 m/s
- 50 m/s

### Snap Lab

#### Converting Potential Energy to Kinetic Energy

In this activity, you will calculate the potential energy of an object and predict the object’s speed when all that potential energy has been converted to kinetic energy. You will then check your prediction.

You will be dropping objects from a height. Be sure to stay a safe distance from the edge. Don’t lean over the railing too far. Make sure that you do not drop objects into an area where people or vehicles pass by. Make sure that dropping objects will not cause damage.

You will need the following:

- Four marbles (or similar small, dense objects)
- Stopwatch

- Metric measuring tape long enough to measure the chosen height
- A scale

Instructions

- Work with a partner. Find and record the mass of four small, dense objects per group.
- Choose a location where the objects can be safely dropped from a height of at least 15 meters. A bridge over water with a safe pedestrian walkway will work well.
- Measure the distance the object will fall.
- Calculate the potential energy of the object before you drop it using
*PE*=*m***g***h*= (9.80)*mh.* - Predict the kinetic energy and velocity of the object when it lands using
*PE*=*KE*and so, $mgh=\frac{m{v}^{2}}{2};\text{}v=\sqrt{2(9.80)h}=4.43\sqrt{h}\text{.}$ - One partner drops the object while the other measures the time it takes to fall.
- Take turns being the dropper and the timer until you have made four measurements.
- Average your drop multiplied by and calculate the velocity of the object when it landed using
**v**=**a***t*=**g***t*= (9.80)*t*. - Compare your results to your prediction.

- Heavy objects do not fall faster than the light objects because while conserving the mechanical energy of the system, the mass term gets cancelled and the velocity is independent of the mass. In real life, the variation in the velocity of the different objects is observed because of the non-zero air resistance.
- Heavy objects do not fall faster than the light objects because while conserving the mechanical energy of the system, the mass term does not get cancelled and the velocity is dependent on the mass. In real life, the variation in the velocity of the different objects is observed because of the non-zero air resistance.
- Heavy objects do not fall faster than the light objects because while conserving the mechanical energy the system, the mass term gets cancelled and the velocity is independent of the mass. In real life, the variation in the velocity of the different objects is observed because of zero air resistance.
- Heavy objects do not fall faster than the light objects because while conserving the mechanical energy of the system, the mass term does not get cancelled and the velocity is dependent on the mass. In real life, the variation in the velocity of the different objects is observed because of zero air resistance.

# Check Your Understanding

### Check Your Understanding

- Kinetic energy is being transformed into potential energy.
- Potential energy is being transformed into kinetic energy.
- Work is being transformed into kinetic energy.
- Kinetic energy is being transformed into work.

True or false—If a rock is thrown into the air, the increase in the height would increase the rock’s kinetic energy, and then the increase in the velocity as it falls to the ground would increase its potential energy.

- True
- False

Identify equivalent terms for *stored energy* and *energy of motion*.

- Stored energy is potential energy, and energy of motion is kinetic energy.
- Energy of motion is potential energy, and stored energy is kinetic energy.
- Stored energy is the potential as well as the kinetic energy of the system.
- Energy of motion is the potential as well as the kinetic energy of the system.