KE, GPE & EPE

When a mass on a vertical spring oscillates up and down, energy is transferred between stores

Although the total energy of the mass-spring system will remain constant, it will have changing amounts of energy in its:
- Elastic potential energy (EPE) store
- Kinetic energy (KE) store
- Gravitational potential energy (GPE) store

Change in Spring Energy, downloadable AS & A Level Physics revision notes

Energy changes when a spring is stretched

At position A:
- The spring has some energy in its elastic potential store since it is slightly compressed
- The spring has zero energy in its kinetic store since it is stationary
- The amount of energy in its gravitational potential store is at a maximum because the mass is at its highest point
At position B:
- The spring has some energy in its elastic potential store since it is slightly stretched
- The energy in its kinetic store is at a maximum as it passes through its resting position at its maximum speed
- The spring has some energy in its gravitational potential store since the mass is still above its lowest point in the oscillation
At position C:
- The energy in the elastic potential store of the spring is at its maximum because it is at its maximum extension
- The spring has zero energy in its kinetic store since it is stationary
- The energy in the gravitational potential store of the spring is at a minimum because it is at its lowest point in the oscillation

Worked example

The diagram below shows a student before and after a bungee jump. The bungee cord has an unstretched length of 30.0 m. KE GPE EPE Worked Example, downloadable AS & A Level Physics revision notes

The mass of the student is 60.0 kg. The gravitational field strength is 9.8 N / kg.

Calculate:

a) The change in gravitational potential energy of the student at 30.0 m

b) The maximum change in the gravitational potential energy of the student

c) The speed of the student after falling 30.0 m if 90% of the energy in the student's gravitational potential store is transferred to the student's kinetic store

d) The spring constant of the bungee cord if all the energy in the gravitational potential store of the student is transferred to the elastic potential store of the bungee cord

Part (a)

Step 1: List the known quantities

- Mass of the student, m = 60.0 kg
- Gravitational field strength, g = 9.8 N/kg
- Change in height, h = 30.0 m

Step 2: Write out the equation for gravitational potential energy

$E_{P} = m g h$

Step 3: Calculate the change in gravitational potential energy

$E_{P} = 60 \times 9.85 \times 30$

$E_{P} = 17 640 J$

Part (b)

Step 1: List the known quantities

- Mass of the student, m = 60.0 kg
- Gravitational field strength, g = 9.8 N/kg
- Maximum change in height, h = 75.0 m

Step 2: Calculate the maximum change in gravitational potential energy

$E_{P m a x} = m g h_{m a x}$

$E_{P m a x} = 60 \times 9.8 \times 75$

$E_{P m a x} = 44 100 J$

Part (c)

Step 1: List the known quantities

- Mass of the student, m = 60.0 kg
- $E_{P}$ at 30.0 m = 17 640 J
Step 2: Determine 90% of the $E_{P}$ at 30.0 m
$E_{K} = 90 % of E_{P}$
$E_{K} = 0.9 \times 17 640$
$E_{K} = 15 876 J$

Step 3: Write out the equation for KE

$E_{K} = \frac{1}{2} m v^{2}$

Step 4: Rearrange to make speed the subject

- Multiply both sides by 2:

$m v^{2} = 2 \times E_{K}$

- Divide both sides by m:

$v^{2} = \frac{2 \times E_{K}}{m}$

- Take the square root of both sides:

$v = \sqrt{\frac{2 \times E_{K}}{m}}$

Step 5: Calculate the speed

$v = \sqrt{\frac{2 \times 15 876}{60}}$

$v = 23.0 m / s$

Part (d)

Step 1: List the known quantities

- $E_{P m a x}$ = 44 100 J
- $E_{e}$ at 75.0 m = $E_{e m a x}$
Step 2: Determine the extension of the bungee cord
$e = 75.0 - 30.0$
$e = 45.0 m$

Step 3: Write out the equation for elastic potential energy

$E_{e} = \frac{1}{2} k e^{2}$

Step 3: Rearrange to make spring constant, k, the subject

- Multiply both sides by 2:

$k e^{2} = 2 \times E_{e}$

- Divide both sides by $e^{2}$ :

$k = \frac{2 \times E_{e}}{e^{2}}$

Step 4: Calculate the spring constant

$k = \frac{2 \times 44 100}{45^{2}}$

$k = 43.6 N / m$

Examiner Tip

If a question asks you to "state" a value, you do not need to carry out a calculation: The answer will almost certainly be a number either from a previous answer or which was given somewhere in the question.

For example, if you have just calculated the gravitational potential energy of an object and are then asked to state the kinetic energy a moment later, the answers are very likely to be the same.

KE, GPE & EPE (AQA GCSE Physics: Combined Science)

Revision Note

KE, GPE & EPE

Worked example

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Author: Leander