Questions: Potential Energy and Conservative Forces

Question 1

A 1 kg ball is dropped straight down from height h = 5 m. A second 1 kg ball rolls down a curved frictionless ramp from the same height. Which ball has more kinetic energy at the bottom?

• A. The dropped ball — it takes a shorter, more direct path and therefore does less work against gravity
• B. The rolled ball — the curved path allows gravity to act over a longer distance, generating more energy
• C. Both have equal kinetic energy — gravity is a conservative force and only the height difference determines the work done
• D. It depends on the exact shape of the ramp curve

Answer: (C)

Gravity is a conservative force: the work it does depends only on the vertical displacement, not on the path taken. Both balls fall the same height h, so gravity does the same work mgh on each, giving both the same final kinetic energy. This is the defining feature of conservative forces — path independence. Options A and B both incorrectly assume that path length or shape affects the energy gained; this would be true for non-conservative forces like friction, but not for gravity.

Question 2

Which of the following forces is NOT conservative?

• A. Gravity near Earth's surface
• B. Spring restoring force (F = -kx)
• C. Kinetic friction
• D. Electrostatic force between fixed charges

Answer: (C)

Kinetic friction is not conservative — the work it does depends on the path length, not just the endpoints. Sliding an object back and forth across a surface dissipates more energy the longer the path, so there is no potential energy function for friction. Gravity, spring forces, and electrostatic forces are all conservative: their work depends only on starting and ending positions, and each has a corresponding potential energy function (U = mgh, U = ½kx², and U = kq₁q₂/r, respectively).

Question 3

The work done by a conservative force depends on the length of the path traveled between two points.

• T. True
• F. False

Answer: False

False — this is the definition of a non-conservative force. A conservative force does work that depends only on the starting and ending positions, not on the path connecting them. This path-independence is what allows us to define a potential energy function: because the work is the same no matter how you travel between two points, we can assign a unique potential energy to each position. If work depended on path length, no such function could exist.

Question 4

When friction acts on a system alongside gravity, the total mechanical energy (KE + PE) is no longer conserved, and the work done by friction must be accounted for separately.

• T. True
• F. False

Answer: True

True. Conservation of mechanical energy (KE + PE = constant) only holds when all forces doing work are conservative. Friction is non-conservative: it converts mechanical energy into heat, reducing the total. The corrected energy equation is ΔE_mechanical = W_non-conservative, where W_non-conservative is negative for friction (energy is lost). Conservative forces still contribute through changes in potential energy; friction appears on the right-hand side as an explicit energy loss. This is why a sliding block doesn't reach the same height on the way back up as it started — friction has dissipated some of the initial mechanical energy.

Question 5

Explain what it means for a force to be conservative, and describe how the relationship F = -dU/dx connects the force to its potential energy function.

Model answer: A force is conservative if the work it does between any two points is independent of the path taken — equivalently, the work done around any closed loop is zero. This property allows us to assign a unique potential energy value U to every position. The connection F = -dU/dx (or F = -∇U in 3D) encodes the physical intuition that force points toward lower potential energy: the negative sign means the force pushes in the direction of decreasing U, just as a ball rolls downhill. For gravity, U = mgh, so F = -d(mgh)/dh = -mg (pointing downward, consistent with gravity). For a spring, U = ½kx², so F = -kx (pointing back toward equilibrium).

The negative sign is critical and often missed. It means conservative forces are 'restoring' in the sense that they drive systems toward lower energy states. This also ensures that potential energy and kinetic energy trade off correctly: as U decreases (ball falls), KE increases by the same amount, keeping E = KE + PE constant.