Lesson 3 Gravitational and Kinetic Energy Calculations

Gravitational potential energy can be found in many natural and technological systems.

C3.2 Logging crane

When work is done against gravity, the force is the weight of the object or the force of gravity working on it. The distance over which work is done against gravity must be the height that you lift the object.

How is the amount of gravitational potential energy in a system calculated?

Gravitational potential energy is determined by the mass of the object, acceleration due to gravity, and the height the object is being lifted against the force of gravity.

E_{p} = m g h E_{p} = g r a v i t a t i o n a l p o t e n t i a l e n e r g y (J) m = m a s s (k g) g = a c c e l e r a t i o n d u e t o g r a v i t y = 9.81 m / s^{2} h = h e i g h t (m)

Remember that work is also a measurement of energy, so its units are joule (J) as well.

Examples

Example 1

A picture with a mass of 3.75 kg hangs on a wall 2.3 m above the floor. What is the gravitational potential energy of the picture relative to the floor?
- Step 1
- Step 2
- Step 3
- Step 4
Step 1: List the variables.

$E_{p} = ? m = 3.75 kg g = 9.81 \frac{m}{s^{2}} h = 2.3 m$

Step 2: Identify the correct formula and rearrange if necessary.

E_p = mgh

Step 3: Substitute the values into the formula.

$E_{p} = (3.75 kg) (9.81 \frac{m}{s^{2}}) (2.3 m)$

Step 4: Calculate the answer.

E_p = 84.611 25 J

The answer must be rounded to two significant digits.

The gravitational potential energy of the picture is 85 J.

Example 2

A roller coaster raises its passengers a vertical distance of 3 650 cm above the ground. The total mass of the roller coaster and passengers is 6.23 × 10³ kg.

What is the gravitational potential energy of the roller coaster and its passengers relative to the ground?
- Step 1
- Step 2
- Step 3
- Step 4
Step 1: List the variables.

$E_{p} = ? m = 6.23 \times 10^{3} kg g = 9.81 \frac{m}{s^{2}} h = 3 650 cm = ? m$

Since h is measured in metres, we have to convert our units of cm to m.

$3 650 cm \times (\frac{1 m}{100 cm}) = 36.5 m$

Step 2: Identify the correct formula and rearrange if necessary.

E_p = mgh

Step 3: Substitute the values into the formula.

$E_{p} = (6.23 \times 10^{3} kg) (9.81 \frac{m}{s^{2}}) (36.5 m)$

Step 4: Calculate the answer.

E_p = 2 230 744.95 J

The answer must be rounded to three significant digits.

2 230 744.95 J cannot be rounded to three significant digits, so it must be put into scientific notation.
Move the decimal point to the left until your answer is between 1 and 10.
2 230 744.95: Move the decimal six places to the left to become 2.230 744 95.
Six decimals to the left is indicated by a 10⁶ (“6” for moving six places to the left).
Round the value of 2.23074495 to 3 sig digs: 2.23.

The gravitational potential energy of the roller coaster and its passengers is 2.23 × 10⁶ J.
How much work did the engine do on the roller coaster and its passengers in carrying them to the top?

Since the work required to lift the roller coaster and its passengers was the force required to raise them to the height against gravity, then the gravitational potential energy and work will be the same.
- Step 1
Step 1:

W = 2.23 × 10 ⁶ J

Since the work required to lift the roller coaster and its passengers was the force required to raise them to the height against gravity, then the gravitational potential energy and work will be the same.

Example 3

How far would you have to lift a 7 . 25 kg bowling ball to give it 242 J of gravitational potential energy?
- Step 1
- Step 2
- Step 3
- Step 4
Step 1: List the variables.

$E_{p} = 242 J m = 7.25 kg g = 9.81 \frac{m}{s^{2}} h = ?$

Step 2: Identify the correct formula and rearrange if necessary.

E_p = mgh

To isolate h, you must divide each side by mg. To move m and g to the other side, you must use the opposite operation.

$\frac{E_{p}}{m g} = \frac{m g h}{m g}$

Now, cancel the like terms.

$\frac{E_{p}}{m g} = h or h = \frac{E_{p}}{m g}$

Step 3: Substitute the values into the formula.

$h = \frac{242 J}{(7.25 kg) (9.81 \frac{m}{s^{2}})}$

Step 4: Calculate the answer.

h = 3.402...m

The answer must be rounded to three significant digits.

The bowling ball would need to be lifted 3.40 m.

Example 4

A child is swinging on a swing in her backyard. She reaches a height of 2.64 m above the ground. If she had 607 J of gravitational potential energy relative to the ground at its highest point, what is the child’s mass?
- Step 1
- Step 2
- Step 3
- Step 4
Step 1: List the variables.

$E_{p} = 607 J m = ? g = 9.81 \frac{m}{s^{2}} h = 2.64 m$

Step 2: Identify the correct formula and rearrange if necessary.

E_p = mgh

To isolate m, you must divide each side by gh. To move g and h to the other side, you must use the opposite operation.

$\frac{E_{p}}{g h} = \frac{m g h}{g h}$

Now, cancel the like terms.

$\frac{E_{p}}{g h} = m or m = \frac{E_{p}}{h g}$

Step 3: Substitute the values into the formula.

$m = \frac{607 J}{(2.64 m) (9.81 \frac{m}{s^{2}})}$

Step 4: Calculate the answer.

m = 23.437...kg

The answer must be rounded to three significant digits.

The child's mass is 23.4 kg.

Please read pages 173 to 175 in your Science 10 textbook. Make sure you take notes on your readings to study from later. You should focus on how the amount of gravitational potential energy can be calculated using the E_p = mgh formula. Remember, if you have any questions or do not understand something, ask your teacher!

Complete the following practice questions to check your understanding of the concept you just learned. Make sure you write complete answers to the practice questions in your notes. After you have checked your answers, make corrections to your responses (where necessary) to study from.

A 57.3 g tennis ball is held above the ground as the tennis player prepares to serve it to his opponent. If he holds the ball at a height of 2.45 m, what is the ball’s gravitational potential energy relative to the ground?

Check your work.

Step 1: List the variables.

E_{p} = ? m = 57.3 g = 0.0573 kg g = 9.81 \frac{m}{s^{2}} h = 2.45 m

Step 2: Identify the correct formula and rearrange if necessary.

E_p = mgh

Step 3: Substitute the values into the formula.

E_{p} = (0.0573 kg) (9.81 \frac{m}{s^{2}}) (2.45 m)

Step 4: Calculate the answer.

E_p = 1.377...J

The answer must be rounded to three significant digits.

The gravitational potential energy of the tennis ball is 1.38 J.

You carry a heavy box filled with books up three flights of stairs for a total vertical height of 10.9 m. If you do 1 450 J of work, what is the mass of the box filled with books?

Check your work.

Step 1: List the variables.

\begin{array}{rcl} E_{p} & = & 1 450 J \\ m & = & ? \\ g & = & 9.81 \frac{m}{s^{2}} \\ h & = & 10.9 m \end{array}

Step 2: Identify the correct formula and rearrange if necessary.

E_p = mgh

To isolate m, you must divide each side by gh. To move g and h to the other side, you must use the opposite operation.

\frac{E_{p}}{g h} = \frac{m g h}{g h}

Now, cancel the like terms.

\frac{E_{p}}{g h} = m or m = \frac{E_{p}}{h g}

Step 3: Substitute the values into the formula.

m = \frac{1 450 J}{(10.9 m) (9.81 \frac{m}{s^{2}})}

Step 4: Calculate the answer.

m = 13.560...kg

The answer must be rounded to three significant digits.

The mass of the box filled with books is 13.6 kg.

Lesson 3 Gravitational and Kinetic Energy Calculations

Gravitational Potential Energy

Gravitational potential energy can be found in many natural and technological systems.

Gravitational potential energy can be found in many natural and technological systems.

Examples

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Practice Questions