AP Physics 1 · Unit 7 · Oscillations

Hooke’s Law and Oscillating Spring Energy Practice Problems

Use restoring force and energy to predict a spring–cart system’s motion at different positions.

12 questions6 conceptual · 6 calculation

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Simulation scene from the Hooke’s Law and Oscillating Spring Energy lesson. — A scene from the related lesson. Open the practice to work through the questions.

What you’ll practice

Connect deformation, force, and energy

Find force and acceleration from position

Use signed displacement from natural length in Fs = −kx, then apply Newton’s second law. Distinguish zero acceleration at equilibrium from zero speed at a turning point.

Calculate speed without a time equation

Use the conserved kinetic-plus-elastic energy to find maximum speed and speeds at compressed or stretched positions. Locate both positions associated with a given speed fraction.

Account for each energy store

Compare kinetic and elastic energy against position, and add gravitational potential energy for a cart attached to a spring on an incline. Keep the height reference consistent.

A question from this set

Restoring force follows deformation

A cart attached to a horizontal spring is left of its natural-length position and is momentarily moving farther left. Which way do its spring force and acceleration point?

Both right, toward equilibrium.
Both left, in the direction of velocity.
Both are zero until the cart reverses.

Try it in the practice set

The cart begins on the compressed side and moves farther left. Identify the restoring direction from deformation.

Key ideas

Follow the energy at each position

Equal deformation magnitudes give equal spring energies

The square in Us = ½kx² makes elastic energy the same at +x and −x. At those two positions a horizontal oscillator has the same speed, while its restoring accelerations point in opposite directions.

Energy-versus-position curves describe the energy balance

Elastic energy curves upward with x², kinetic energy curves downward within the allowed positions, and their sum remains constant. These plots compare energy at locations; sinusoidal motion-versus-time graphs are outside this set.

See the physics

Compare real spring motion with energy bars

After an attempt, release an attached cart under native spring forces and inspect measured crossings and turning points. Compare both sides of equilibrium, change the release distance, and follow an inclined cart through equally spaced markers. Energy bars and energy-versus-position plots reveal how the energy is shared.

Use a hint for a starting point or open a worked solution to follow the reasoning. Your progress saves in this browser, so you can return to unfinished questions.

Ready for hooke’s law and oscillating spring energy?

Begin with a conceptual question, then work through the calculations.

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