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

1.7 Energy – Gravitational Potential Energy (GPE)

Learning objective

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

$$\Delta E_{p}=m\,g\,\Delta h$$

and relate it to the broader energy concepts required by the Cambridge IGCSE Physics (0625) syllabus.

How the topic fits into the syllabus

Syllabus code	Topic	Relevance to GPE
1.7.1	Energy stores (core & supplement)	GPE is one of six energy stores; see the overview table below.
1.7.2	Work	Work done against gravity stores energy as GPE ( W = F·d = m g Δh ).
1.7.3	Power	Power required to lift a mass at constant speed: P = ΔEₚ / t.
1.7.4	Efficiency	Real lifts (e.g., pulleys) have input work > ΔEₚ; efficiency = ΔEₚ / input × 100 %.
1.5 (supplement)	Momentum & vectors	When GPE is converted to kinetic energy, momentum is conserved (e.g., a falling object).
1.5 (supplement)	Scalars vs vectors	GPE is a scalar; direction is handled by the sign of Δh.
1.5 (supplement)	Turning effect of forces	Compare GPE with elastic potential energy (EPE = ½ k x²) to illustrate different energy stores.
2‑5 (core)	Energy transfer mechanisms	GPE can be transformed into kinetic, thermal, electrical (hydro‑electric plant) or wave energy.

Energy‑store overview (core)

Store	Symbol	Typical example
Kinetic energy (KE)	K	Moving car
Gravitational potential energy (GPE)	Eₚ	Book on a shelf
Elastic potential energy (EPE)	Eₑ	Stretched spring
Chemical energy	–	Battery, food
Nuclear energy	–	Uranium fuel
Thermal energy	–	Hot water

Key concepts for gravitational potential energy

• ΔEₚ (J) – change in GPE when an object moves vertically in a uniform gravitational field.
• m (kg) – mass of the object.
• g (N kg⁻¹ = m s⁻²) – gravitational field strength; on Earth ≈ 9.8 m s⁻².
• Δh (m) – vertical displacement; upward = positive, downward = negative.

Units table

Quantity	Symbol	SI unit
Mass	m	kg
Gravitational field strength	g	N kg⁻¹ (= m s⁻²)
Height change	Δh	m
GPE change	ΔEₚ	J (joule)

Formula, rearrangements & sign conventions

Primary equation

$$\Delta E_{p}=m\,g\,\Delta h$$

Re‑arranged forms (useful for exam questions)

• Mass: \(m=\dfrac{\Delta E_{p}}{g\,\Delta h}\)
• Height change: \(\Delta h=\dfrac{\Delta E_{p}}{m\,g}\)
• Field strength: \(g=\dfrac{\Delta E_{p}}{m\,\Delta h}\)

Sign of Δh

• Δh > 0 (object rises) → ΔEₚ > 0 (energy stored).
• Δh < 0 (object falls) → ΔEₚ < 0 (energy released, usually as kinetic energy).

Derivation (brief)

The work done against gravity to move an object vertically a distance Δh is

$$W = F\;d = (m g)\,\Delta h = m g \Delta h$$

Because the work done on the object is stored as gravitational potential energy, W = ΔEₚ. This links the concepts of work (1.7.2) and GPE directly.

Link to related syllabus topics

• Momentum (1.5, supplement): When a body falls, the loss of GPE becomes kinetic energy; the resulting speed can be used together with mass to calculate momentum \(p = m v\). Momentum is conserved in collisions, even though the energy form may change.
• Elastic potential energy: Both GPE and EPE are scalar energy stores that can inter‑convert (e.g., a roller‑coaster hill → spring‑loaded launch).
• Power (1.7.3): Lifting a mass at constant speed requires a power P = ΔEₚ / t. This connects the rate of energy transfer to the GPE formula.
• Efficiency (1.7.4): Real devices (e.g., pulley systems) need more input work than the ideal ΔEₚ. Efficiency = (ΔEₚ / input work) × 100 %.
• Energy‑transfer mechanisms (core): GPE → kinetic → thermal (via friction) → electrical (hydro‑electric turbine) → wave energy (water waves).

Step‑by‑step procedure for solving GPE problems

1. Read the question carefully; identify the unknown quantity.
2. Write down all given values, converting to SI units (g → kg, cm → m, etc.).
3. Choose the appropriate form of the equation (solve for ΔEₚ, m, Δh or g).
4. Insert the numerical values, keeping the sign of Δh consistent with the direction described.
5. Carry out the calculation, retaining at least three significant figures.
6. State the final answer with the correct unit (J) and sign (+ increase, – decrease).

Worked examples

Example 1 – Positive height change (basic GPE)

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. Given: \(m = 2.5\ \text{kg}\), \(\Delta h = +1.2\ \text{m}\), \(g = 9.8\ \text{N kg}^{-1}\).
2. \(\Delta E_{p}=m g \Delta h = (2.5)(9.8)(1.2)= 29.4\ \text{J}\).
3. Answer: **+29.4 J** (GPE increases).

Example 2 – Negative height change (fall)

Question: A 0.8 kg ball is dropped from a height of 5.0 m. What is the change in its gravitational potential energy just before it hits the ground?

Solution:

1. Take upward as positive, so \(\Delta h = -5.0\ \text{m}\).
2. \(\Delta E_{p}= (0.8)(9.8)(-5.0)= -39.2\ \text{J}\).
3. Interpretation: the ball loses **39 J** of GPE, which is converted into kinetic energy (and later into heat).

Example 3 – Solving for mass

Question: An object gains 150 J of GPE when lifted 3 m. Find its mass.

Solution:

1. Re‑arranged formula: \(m = \dfrac{\Delta E_{p}}{g\,\Delta h}\).
2. \(m = \dfrac{150}{9.8 \times 3}=5.10\ \text{kg}\) (3 sf).

Example 4 – Power required to lift a mass

Question: How much power is needed to lift a 10 kg sack to a height of 2 m in 4 s at constant speed?

Solution:

1. ΔEₚ = m g Δh = (10)(9.8)(2) = 196 J.
2. Power = ΔEₚ / t = 196 J / 4 s = 49 W.

Example 5 – Efficiency of a simple pulley lift

Question: A student uses a hand‑pulley to raise a 5 kg bucket 0.8 m. The spring‑scale reads an average force of 55 N. Calculate the efficiency of the lift.

Solution:

1. Input work = F × d = 55 N × 0.8 m = 44 J.
2. Ideal GPE gain = m g Δh = (5)(9.8)(0.8) = 39.2 J.
3. Efficiency = (39.2 J / 44 J) × 100 % ≈ 89 %.

Practical tip & safety (AO3)

• Measuring m and Δh: Use a calibrated balance (±0.01 kg) and a metre‑stick or motion sensor (±0.01 m). Record values with the appropriate number of significant figures.
• Apparatus safety: Secure the object to a string or platform before lifting. Keep the work area clear, and wear safety goggles in case the load falls.
• Experimental verification: Compare the calculated ΔEₚ with the work measured by a spring‑scale (force × distance). The two should agree within experimental uncertainty, confirming energy conservation.

Common mistakes to avoid

• Using grams or centimetres instead of kilograms and metres – SI units are mandatory.
• Assigning the wrong sign to Δh; remember upward = +, downward = –.
• Confusing the symbol g (field strength, N kg⁻¹) with the acceleration 9.8 m s⁻² – they have the same numerical value on Earth but different unit expressions.
• Omitting the joule (J) in the final answer or giving an incorrect number of significant figures.
• Treating GPE as a vector; it is a scalar, and direction is handled by the sign of Δh.

Extension questions (challenge yourself)

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.
4. Calculate the mass of an object that loses 250 J of GPE when it falls 4 m.
5. On a planet where \(g = 3.7\ \text{N kg}^{-1}\), how much GPE does a 5 kg rock gain when lifted 2 m?
6. Using the power formula, determine the minimum power a motor must supply to raise a 20 kg load 3 m in 5 s at constant speed.
7. Discuss how the GPE lost by water in a hydro‑electric dam is ultimately converted into electrical energy, linking to the “energy‑transfer mechanisms” part of the syllabus.

Summary

The change in gravitational potential energy for an object moving vertically in a uniform gravitational field is

$$\Delta E_{p}=m\,g\,\Delta h$$

Key points to remember:

• All quantities must be expressed in SI units (kg, m, s⁻², J).
• Δh > 0 → ΔEₚ > 0 (energy stored); Δh < 0 → ΔEₚ < 0 (energy released).
• The formula can be rearranged to find any one of the three variables (m, g, Δh).
• GPE is a scalar energy store that inter‑converts with kinetic, elastic, thermal, and electrical forms, linking the whole IGCSE energy topic.
• Work, power and efficiency provide the practical framework for measuring and applying the GPE concept in real‑world situations.