Potential Energy at a Height – Long Answer Questions
Medium Level (Application & Explanation)
Q1. Define gravitational potential energy and derive the formula using the idea of work done against gravity.
Answer:
Gravitational potential energy (PE) is the energy stored due to height above a chosen
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level.
It equals the work done to lift the object slowly against gravity.
For a mass m, weight is mg (downward).
To lift at constant speed, the upward force must be mg.
Work done = Force × displacement = mg × h.
So, PE = mgh. Units: Joule (J). Here, m in kg, g in m/s², h in m.
Q2. A 12 kg box is lifted to 5 m. Calculate its potential energy and explain each step and unit used.
Answer:
Given: m = 12 kg, h = 5 m. Take g = 9.8 m/s².
Formula: PE = mgh.
Substitute: PE = 12 × 9.8 × 5.
Compute: PE = 588 J.
The unit Joule comes from kg·m²/s².
This energy is stored due to position. It can convert to kinetic energy when the box falls.
Q3. Explain why the work done in lifting a block depends only on vertical height and not on the path taken.
Answer:
PE depends on the change in height (h) only.
If you use a ramp or lift it straight up, the vertical rise is what matters.
Work done against gravity = mg × vertical height.
The path may be longer, but gravity acts vertically.
So the extra horizontal distance does not add work against gravity.
Hence, PE = mgh is the same for all paths (ignoring friction).
Q4. How does changing the
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level affect potential energy? Explain with a small example.
Answer:
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level is where we take h = 0.
Changing it shifts the numerical value of PE.
But the change in PE between two positions stays the same.
Example: m = 2 kg, g = 10 m/s². From 1 m to 3 m above ground.
Change in PE = mg(3 − 1) = 2 × 10 × 2 = 40 J.
If the table top is new zero, heights change, but the 40 J change remains.
Q5. Show how PE changes when mass or height is doubled or halved. Use simple numbers to illustrate.
Answer:
PE = mgh shows PE is directly proportional to m and h.
If you double mass, PE doubles.
If you double height, PE doubles.
If you halve mass, PE becomes half.
Example: m = 3 kg, h = 4 m, g = 10 m/s² → PE = 120 J.
Double m to 6 kg → 240 J; double h to 8 m → 240 J; double both → 480 J.
High Complexity (Analysis & Scenario-Based)
Q6. A worker lifts a 15 kg bag from the ground to 2.0 m in two steps: 0 to 1.2 m, then 1.2 m to 2.0 m. Compare the work done in each step and the total work with a single direct lift.
Answer:
Take g = 10 m/s² for easy math.
Step 1: Δh = 1.2 m → W₁ = mgΔh = 15 × 10 × 1.2 = 180 J.
Step 2: Δh = 0.8 m → W₂ = 15 × 10 × 0.8 = 120 J.
Total work in steps = W₁ + W₂ = 300 J.
Direct lift to 2.0 m: W = mgh = 15 × 10 × 2 = 300 J.
Same result. Work depends on height change only, not on the path (ignoring friction).
Q7. Two objects are placed at different heights: A is 8 kg at 6 m, B is 12 kg at 4 m. Whose potential energy is greater? Explain.
Answer:
Use PE = mgh, with the same g for both.
For A: PE_A = 8 × g × 6 = 48g.
For B: PE_B = 12 × g × 4 = 48g.
So, PE_A = PE_B. Both have the same PE.
The product m × h is the same for both (48).
Conclusion: With the same g, equal m × h gives equal potential energy.
Q8. A 5 kg stone is dropped from 10 m. Find the PE at the top, the PE after falling 6 m, and explain energy changes as it falls.
Answer:
Take g = 10 m/s² for easy math.
At 10 m: PE_top = mgh = 5 × 10 × 10 = 500 J.
After falling 6 m, it is at 4 m: PE = 5 × 10 × 4 = 200 J.
PE lost = 500 − 200 = 300 J.
This lost PE becomes kinetic energy (KE) (ignoring air resistance).
As it falls, PE decreases and KE increases. Total mechanical energy stays constant.
Q9. Two students use different zero levels. A takes the ground as zero. B takes a table 0.8 m high as zero. A 3 kg book is moved from 1.0 m (on the table) to a shelf at 1.5 m. Compare their PE calculations and the change in PE.
Answer:
Take g = 10 m/s².
Student A (ground zero): Initial h = 1.0 m, Final h = 1.5 m.
Student B (table zero): Initial h = 0.2 m? Wait—table is at 0.8 m, book is on table at 1.0 m → relative h = 0.2 m; shelf at 1.5 m → relative h = 0.7 m.
Change (B) = 3 × 10 × (0.7 − 0.2) = 15 J. Different absolute PEs, but same change in PE and same work.
Q10. A crane lifts Load X (500 kg) to 8 m and Load Y (250 kg) to 16 m. Which job needs more energy? What principle explains your answer?
Answer:
Use PE = mgh with the same g.
For X: PE_X = 500 × g × 8 = 4000g.
For Y: PE_Y = 250 × g × 16 = 4000g.
Both jobs need the same energy.
The product m × h is the same, so PE gained is equal.
Principle: Work done against gravity depends only on mgh, not on how mass and height are split.
