Quick answer: Work done by a variable force is the integral W = ∫F dx, the area under the force–position graph. This AP Physics C worksheet on work energy theorem gives 20 problems from easy to hard: constant force work, units of work and stretching a spring. The worked answers are on a separate page, and the worksheet is a free printable PDF.
These problems build the habit of writing work as an integral, reading energy from graphs, and moving between force and potential energy by differentiation. They start with constant forces and springs, then move to variable forces, equilibrium, and potential curves. Level: AP Physics C: Mechanics unit 3 and first-year university.
Level: AP Physics C (Mechanics and E&M, calculus-based) · Curriculum: AP PC:M Unit 3
Problem 1 Constant force work ★★★
A dock worker pulls a crate with a 40 N rope force through 5.0 m along level ground.1. Calculate the work if the rope is parallel to the ground.
2. Then calculate the work if the rope makes 60° with the ground.
Problem 2 Units of work ★★★
1. Show that the area under an F–x graph has the unit of joule.2. Express the joule in base SI units.
Problem 3 Stretching a spring ★★★
Emma in Leeds stretches a spring with k = 200 N/m by 0.30 m from its natural length.1. Calculate the work she does.
Problem 4 Work from rest ★★★
A net work of 120 J is done on a 3.0 kg trolley that starts at rest.1. Calculate its final speed.
Problem 5 Force from a spring potential ★★★
The potential energy of a spring is U = ½kx² with k = 50 N/m.1. Use F = −dU/dx to find the force.
2. Then calculate the force at x = 0.20 m.
Problem 6 Conservative force ★★★
1. Define a conservative force.2. Then state whether kinetic friction is conservative and give a reason.
Problem 7 Linear force ★★★
A force F = 4.0x newtons acts from x = 0 to x = 2.5 m.1. Calculate the work by integration.
2. Then check using the area of a triangle.
Problem 8 Force with an offset ★★★
Kofi in Accra pulls a 1.5 kg sled from rest with F = (5.0 + 2.0x) N over 0 to 4.0 m. Ignore friction.1. Calculate the work.
2. Then calculate the final speed.
Problem 9 Piecewise force graph ★★★
The graph shows the force on a trolley as it moves along a track.2. Then calculate the work from 2 m to 5 m.
3. Finally, calculate the total work from 0 to 6 m.
Problem 10 Force from a cubic potential ★★★
A particle has U(x) = 2.0x³ − 6.0x in joules, x in metres.1. Find F(x).
2. Then calculate F at x = 2.0 m.
3. Finally, find the equilibrium positions and classify them.
Problem 11 Motorbike power ★★★
A motorbike travels at a steady 20 m/s against a total resistive force of 600 N.1. Calculate the power of the engine.
2. Then calculate the work done in 30 s.
Problem 12 Gravity from potential ★★★
The gravitational potential energy of a mass m at distance r from Earth’s centre is U = −GMm/r, with GM = 3.99 × 10¹⁴ m³/s².1. Find F(r).
2. Then calculate the force on 1.0 kg at r = 6.37 × 10⁶ m.
Problem 13 Average force mistake ★★★
Bilal in Lahore says: “The work of F = 3.0x² from 0 to 2.0 m is the average force times the distance, so (0 + 12)/2 × 2.0 = 12 J.”1. Identify the error.
2. Then calculate the correct work.
Problem 14 Reading an energy diagram ★★★
A 0.50 kg particle moves in the potential U = 2.0x² J with total energy E = 8.0 J.2. Then calculate the maximum speed.
3. Finally, state where the maximum speed occurs.
Problem 15 Lifting out of the Earth’s field ★★★
Mei in Singapore asks how much work is needed to lift a 1.0 kg mass from Earth’s surface (R = 6.37 × 10⁶ m) to a height equal to R. Use GM = 3.99 × 10¹⁴ m³/s².1. Write W as an integral and evaluate it.
2. Then compare with mgh using g = 9.8 m/s² and h = R.
Problem 16 Escape speed ★★★
Using the same data, derive and calculate the speed needed for a projectile to escape Earth from the surface, ignoring air.1. Find the work to take 1 kg to infinity.
2. Equate with kinetic energy to find v.
Problem 17 Molecular potential ★★★
The interaction of two atoms is modelled by U(r) = U₀[(a/r)¹² − 2(a/r)⁶], with U₀ = 1.0 × 10⁻²¹ J and a = 3.0 × 10⁻¹⁰ m.1. Show that r = a is an equilibrium separation and give U there.
2. Then calculate the effective spring constant for small oscillations, k = d²U/dr² at r = a.
Problem 18 Sliding in a double well ★★★
A 1.0 kg bead moves on a wire with potential U = x⁴ − 4.0x² joules (x in metres). It is released from rest at x = 0.50 m.1. Calculate F at the release point and the initial direction.
2. Then calculate its speed when it reaches x = +√2.
Problem 19 Lifting a chain ★★★
A uniform chain of mass 3.0 kg and length 2.0 m hangs vertically from a table edge. Wanjiru lifts it slowly onto the table.2. Then write the work to lift the element at depth y by y.
3. Integrate to find the total work.
Problem 20 Force changing in time ★★★
A 2.0 kg particle starts from rest and feels F = 6.0t newtons along x.1. Find v(t) by integrating F/m.
2. Then find the power P(t) = Fv.
3. Integrate the power from 0 to 2.0 s and compare with the kinetic energy gained.
