5 AP Physics 1 Spring Problems | TI-84 Calculator Program

Martin McSweeney

Spring problems show up every year on the AP Physics 1 exam — and they cover a lot of ground. Hooke's Law, elastic potential energy, period, and simple harmonic motion speed questions all look different on the surface, but they come from the same small set of formulas.

In this post (and the video above), I walk through 5 of those problems using the MECH2 program for the TI-84 Plus CE. Each one shows the TI-84 input, the result, and the equation worked out by hand so you can see how they match.

The program is part of the Physics Mechanics Bundle — and also included in the AP Physics Ultimate Bundle and the AP STEM Mega Bundle.

Problem 1: Hooke's Law — F = kx

A spring has a spring constant of 240 N/m. It is stretched 0.15 m from equilibrium. What is the magnitude of the spring force?

TI-84 path: Mechanics 2 → Springs → Hooke's Law → F = kx

Inputs:

  • k = 240
  • x = 0.15

Result: F = 36 N

By hand:

F = kx
F = (240)(0.15)
F = 36 N

This is about as straightforward as Hooke's Law gets. If you know the spring constant and the displacement, you have the force.

Problem 2: Elastic Potential Energy — U = (1/2)kx²

A spring with spring constant 180 N/m is compressed 0.20 m. How much elastic potential energy is stored in the spring?

TI-84 path: Springs → Elastic PE → U = (1/2)kx²

Inputs:

  • k = 180
  • x = 0.20

Result: U = 3.6 J

By hand:

U = (1/2)kx²
U = (1/2)(180)(0.20)²
U = (1/2)(180)(0.04)
U = 3.6 J

The classic mix-up: Hooke's Law uses x, but spring potential energy uses x². One formula is linear, the other is quadratic. Worth knowing cold before exam day.

Problem 3: Period of a Mass-Spring System — T = 2π√(m/k)

A 0.50 kg block oscillates on a spring with spring constant 125 N/m. What is the period of oscillation?

TI-84 path: Springs → Period & Freq → T = 2π√(m/k)

Inputs:

  • m = 0.50
  • k = 125

Result: T = 0.40 s

By hand:

T = 2π√(m/k)
T = 2π√(0.50/125)
T = 2π√(0.004)
T ≈ 2π(0.0632)
T ≈ 0.40 s

Note that period depends on mass and spring constant only — not amplitude. A common distractor on the AP exam is changing the amplitude and asking if the period changes. It doesn't.

Problem 4: Maximum Speed in SHM — vmax = A√(k/m)

A block of mass 0.40 kg oscillates on a spring with k = 90 N/m and amplitude A = 0.25 m. What is the maximum speed?

TI-84 path: Springs → Energy → Max velocity

Inputs:

  • A = 0.25
  • k = 90
  • m = 0.40

Result: vmax = 3.75 m/s

By hand:

vmax = A√(k/m)
vmax = 0.25√(90/0.40)
vmax = 0.25√225
vmax = 0.25(15)
vmax = 3.75 m/s

Maximum speed occurs at equilibrium — where x = 0 and all the potential energy has converted to kinetic. The farther the amplitude, the faster the maximum speed.

Problem 5: Speed at a Particular Position — v = √[(k/m)(A² − x²)]

A 0.50 kg block oscillates on a spring with k = 200 N/m and amplitude A = 0.10 m. What is the speed when the block is at x = 0.06 m?

TI-84 path: Springs → Energy → v at position x

Inputs:

  • k = 200
  • m = 0.50
  • A = 0.10
  • x = 0.06

Result: v = 1.60 m/s

By hand:

v = √[(k/m)(A² − x²)]
v = √[(200/0.50)(0.10² − 0.06²)]
v = √[(400)(0.01 − 0.0036)]
v = √[(400)(0.0064)]
v = √2.56
v = 1.60 m/s

This is the most AP-style problem of the five. You're not at equilibrium, you're not at max displacement — you're somewhere in between, and the formula handles it cleanly. The object moves fastest near x = 0 and slows as it approaches maximum displacement (x = A).

Summary

Here's a quick reference for the five formulas covered:

Topic Formula
Hooke's Law F = kx
Elastic Potential Energy U = (1/2)kx²
Period T = 2π√(m/k)
Maximum Speed vmax = A√(k/m)
Speed at Position x v = √[(k/m)(A² − x²)]

If spring problems feel like a lot to keep track of, that's because they are. Five formulas, multiple variables, and it's easy to grab the wrong one under time pressure. Having the program handle the arithmetic lets you focus on which formula applies and why.

Get the Program

The MECH2 program is part of the Physics Mechanics Bundle for the TI-84 Plus CE. It also covers rotational motion, momentum, fluids, and more — everything you need for AP Physics 1 mechanics.

For broader exam coverage:

New to TI-84 programs? Start with the Complete Install Guide (2026) or watch the install and troubleshoot video.

Questions about whether programs are allowed on your exam? See: Are TI-84 programs legal on AP exams?

AP® is a trademark registered by the College Board, which is not affiliated with and does not endorse this product.

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