Recall and use the equation for the change in gravitational potential energy ΔE_p = m g Δh

1.7.1 Energy – Gravitational Potential Energy

Learning Objective

Recall and use the equation for the change in gravitational potential energy:

\$\Delta E_p = m g \Delta h\$

Key Concepts

• Gravitational potential energy (E_p) – energy stored due to an object's position in a gravitational field.
• Mass (m) – amount of matter, measured in kilograms (kg).
• Gravitational field strength (g) – acceleration due to gravity, ≈ \$9.8\ \text{N kg}^{-1}\$ on Earth.
• Height change (Δh) – vertical displacement, measured in metres (m).

Units Table

Derivation (Brief)

The work done against gravity to lift an object a height Δh is:

\$W = F \times s = (mg) \times \Delta h = mg\Delta h\$

Since the work done is stored as gravitational potential energy, \$W = \Delta E_p\$.

Step‑by‑Step Procedure for Solving Problems

1. Identify the mass \$m\$ of the object (kg).
2. Determine the height change \$\Delta h\$ (m). Use a positive value for a rise, negative for a fall.
3. Use \$g = 9.8\ \text{N kg}^{-1}\$ unless another value is given.
4. Substitute the values into \$\Delta E_p = m g \Delta h\$.
5. Calculate the result, keeping track of sign (positive = increase in \$E_p\$, negative = decrease).
6. State the answer with the correct unit (J).

Worked Example

Question: A 2.5 kg textbook is lifted from the floor to a shelf 1.2 m above the floor. Calculate the increase in its gravitational potential energy.

Solution:

1. Mass \$m = 2.5\ \text{kg}\$.
2. Height change \$\Delta h = 1.2\ \text{m}\$ (positive, because the book is raised).
3. Use \$g = 9.8\ \text{N kg}^{-1}\$.
4. Substitute: \$\Delta E_p = (2.5)(9.8)(1.2)\$.
5. Calculate: \$\Delta E_p = 2.5 \times 9.8 = 24.5\$; \$24.5 \times 1.2 = 29.4\ \text{J}\$.
6. Answer: The textbook’s gravitational potential energy increases by \$29.4\ \text{J}\$.

Common Mistakes to Avoid

• Forgetting to convert mass from grams to kilograms.
• Using the wrong sign for \$\Delta h\$ (downward motion should give a negative \$\Delta h\$).
• Mixing up \$g\$ (9.8 N kg⁻¹) with \$g = 9.81\ \text{m s}^{-2}\$ – both are equivalent, but keep units consistent.
• Omitting the unit joule (J) in the final answer.

Extension Questions

1. If a 0.8 kg ball is dropped from a height of 5 m, what is the change in its gravitational potential energy just before it hits the ground?
2. A 12 kg crate is lifted 0.5 m onto a platform. How much work must be done against gravity?
3. Explain why the gravitational potential energy of an object at the top of a hill is greater than at the bottom, even if the object's speed is the same at both points.

Summary

The change in gravitational potential energy of an object moving vertically in a uniform gravitational field is given by \$\Delta Ep = m g \Delta h\$. Remember to keep units consistent, use the correct sign for \$\Delta h\$, and interpret the sign of \$\Delta Ep\$ as an increase (positive) or decrease (negative) in stored energy.