AP® Physics 1: Algebra-Based review sheet from Aim for Five (aimforfive.com/physics/units/7/7-1)
Unit 7 · Topic 7.1
7.1 Defining Simple Harmonic Motion (SHM)
Simple harmonic motion (SHM) is the back-and-forth motion you get when a restoring force pulls an object toward equilibrium with a strength proportional to how far it has been displaced. A block on an ideal spring is the model case, and a pendulum swinging through small angles comes very close.
Key terms
- simple harmonic motion
- restoring force
- equilibrium position
- Hooke's law
- simple pendulum
Periodic motion and the special case of SHM
Periodic motion is any motion that repeats the same pattern over and over: a swing, a heartbeat, a bouncing ball. Simple harmonic motion is one special kind of periodic motion, with a very specific cause.
Every oscillator has an equilibrium position, the spot where the net force on the object is zero. Move the object away from it and a restoring force appears. A restoring force always points opposite the displacement, back toward equilibrium.
The motion is simple harmonic when the size of that restoring force is directly proportional to the displacement: twice as far from equilibrium means twice the pull back. That one condition is what produces the smooth sine-shaped motion you'll study in 7.3.
The spring model
An ideal spring obeys Hooke's law (from 2.8): F = −kx. Here x is the displacement from equilibrium, k is the spring constant in N/m, and the minus sign says the force points opposite the displacement. Stretch the spring to the right and it pulls left; squeeze it and it pushes right.
Put that force into Newton's second law and you get a = −(k/m)x. The acceleration is proportional to the displacement and points the opposite way. That relationship is the fingerprint of SHM. If you see it, or a force graph that is a straight line through the origin with a negative slope, the motion is simple harmonic.
Vertical springs
Hang a mass m on a spring and it settles lower, at a new equilibrium where the spring force balances the weight: kd = mg, so the stretch is d = mg/k.
Measure displacement from that new hanging position and the net force (spring plus gravity) is still −kx, with the same k. So a vertical spring oscillates in SHM about the hanging equilibrium point, not about the spring's unstretched length. Gravity shifts the center of the motion but doesn't change its character.
The pendulum: SHM only for small angles
A simple pendulum is a small mass (the bob) on a light string of length L. When it's pulled aside by an angle θ, gravity exerts a restoring torque about the pivot of size mgL sin θ, pointing back toward the bottom.
That torque is proportional to sin θ, not to θ, so a pendulum is not perfectly simple harmonic. But for small angles, sin θ is almost exactly equal to θ measured in radians. The restoring torque is then very nearly proportional to the angle, and the pendulum behaves as SHM. The table shows how good the approximation is.
| Angle | θ (rad) | sin θ | Error |
|---|---|---|---|
| 5° | 0.0873 | 0.0872 | 0.1% |
| 10° | 0.1745 | 0.1736 | 0.5% |
| 20° | 0.3491 | 0.3420 | 2% |
| 45° | 0.7854 | 0.7071 | 11% |
How to tell whether motion is SHM
- Force (or acceleration) against displacement is a straight line through the origin with a negative slope.
- The restoring force always points toward one equilibrium position.
- The period stays the same when you change the amplitude (see 7.2 and 7.3).
- A ball bouncing on the floor fails these tests: while it's in the air the only force is its constant weight, which doesn't depend on its height. It's periodic, but not simple harmonic.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Force and acceleration on a spring
A 0.50 kg block on a frictionless table is attached to a horizontal spring with k = 200 N/m. Take +x to the right of equilibrium. Find the force on the block and its acceleration when it is (a) 0.040 m to the right of equilibrium and (b) 0.020 m to the left of equilibrium.
Show the solutionHide the solution
- Step 1: (a) x = +0.040 m. F = −kx = −(200 N/m)(0.040 m) = −8.0 N, so 8.0 N to the left.
- Step 2: a = F/m = −8.0 N / 0.50 kg = −16 m/s², so 16 m/s² to the left, back toward equilibrium.
- Step 3: (b) x = −0.020 m. F = −(200)(−0.020) = +4.0 N, so 4.0 N to the right. a = 4.0/0.50 = +8.0 m/s² to the right.
- Step 4: Half the displacement gives half the force and half the acceleration, and both always point toward equilibrium. That's SHM.
Answer: (a) 8.0 N and 16 m/s², both to the left (b) 4.0 N and 8.0 m/s², both to the right
- Example 2Calculator allowed
A vertical spring
A 0.30 kg mass hangs at rest from a spring with k = 60 N/m. Use g = 9.8 m/s². (a) How far is the spring stretched? (b) The mass is pulled 0.030 m below that resting position and held. What is the net force on it?
Show the solutionHide the solution
- Step 1: (a) At equilibrium the spring force balances the weight: kd = mg, so d = mg/k = (0.30)(9.8)/60 = 0.049 m.
- Step 2: (b) Total stretch is now 0.049 + 0.030 = 0.079 m. Spring force = (60)(0.079) = 4.74 N up. Weight = (0.30)(9.8) = 2.94 N down.
- Step 3: Net force = 4.74 − 2.94 = 1.80 N up.
- Step 4: Shortcut: measured from the hanging equilibrium, the net force is just kx = (60)(0.030) = 1.8 N, pointing back toward equilibrium. Gravity only moved the center point.
Answer: (a) 0.049 m (about 4.9 cm) (b) 1.8 N upward
- Example 3Calculator allowed
Is a bouncing ball in SHM? (classic trap)
A rubber ball is dropped and bounces again and again to nearly the same height, so its motion repeats. A student says this is simple harmonic motion. Do you agree? Justify your answer.
Show the solutionHide the solution
- Step 1: Check the defining condition: is the restoring force proportional to the displacement from an equilibrium position?
- Step 2: While the ball is in the air, the only force on it is gravity, mg downward. That force is the same whether the ball is 1 cm or 1 m above the floor. It doesn't grow with distance from any equilibrium point.
- Step 3: The large force from the floor acts only during the brief contact, so it doesn't fit the pattern either.
- Step 4: So the motion is periodic but not simple harmonic. A sign of this: drop it from higher up and the time between bounces gets longer, while a true SHM period doesn't depend on amplitude.
Answer: Disagree. The motion repeats, but the force on the ball isn't proportional to its displacement, so it isn't SHM.
Common mistakes
- Calling any repeating motion SHM. It's SHM only if the restoring force is proportional to the displacement from equilibrium.
- Measuring a vertical spring's displacement from its unstretched length. For SHM, measure from the hanging equilibrium position.
- Using degrees in the small-angle idea. sin θ ≈ θ only works with θ in radians.
- Thinking the restoring force is zero at the turning points because the object is momentarily at rest. The force (and acceleration) is largest there.
On the exam
- You may be asked to decide from a force–displacement graph or data table whether a system is in SHM. Look for a straight line through the origin with a negative slope, and say so in words.
- When justifying why a pendulum is close to SHM, mention that the restoring torque is proportional to sin θ, which is approximately θ for small angles.
Connected topics
Videos
Check yourself
4 questions on 7.1 Defining Simple Harmonic Motion (SHM). Pick an answer to see if you got it, and why.
An object moves along the x-axis, with x = 0 as its equilibrium position. Which of the following net force laws would produce simple harmonic motion? (k is a positive constant.)
A block on a horizontal ideal spring is pulled to x = +0.10 m, where x = 0 is the equilibrium position, and released from rest. At the instant of release, which describes the net force on the block?
A 0.50 kg block hangs from a vertical ideal spring with spring constant 50 N/m. How far is the spring stretched when the block hangs at its equilibrium position? Use g = 10 m/s².
Why can a simple pendulum swinging through small angles be modeled as simple harmonic motion?
0 of 4 answered