Grade by Grade
C. Energy and Momentum · Work, Energy and Simple Harmonic Motion

Simple Harmonic Motion of a Mass on a Spring

C3.2 C3.1 C2.1About 70 minutes
Learning goals
  • I can describe simple harmonic motion and explain how it follows from Hooke's law.
  • I can use amplitude, period and frequency to describe an oscillation.
  • I can calculate the period of a mass on a spring with T = 2π√(m/k), and find k or m from a period.
  • I can describe where the speed and acceleration are greatest and smallest during SHM.

Words to know

simple harmonic motion (SHM)
Periodic motion in which the restoring force, and so the acceleration, is proportional to the displacement from equilibrium and opposite to it.
amplitude (A)
The maximum displacement from the equilibrium position.
period (T)
The time for one complete cycle of an oscillation, in seconds.
frequency (f)
The number of cycles per second, in hertz (Hz); f = 1/T.
oscillation
A repeated back-and-forth motion about an equilibrium position.

Back and forth, again and again

Pull a mass on a spring down and let go. It bobs up and down, passing the same point again and again in the same time. A playground swing, a guitar string, a car bouncing on its springs after a bump, the atoms in a solid: all of these repeat a motion over and over. The simplest and most important repeating motion is simple harmonic motion (SHM), and the mass on a spring is its model example.

Remember from Grade 11: in SPH3U's Waves and Sound unit you used period (T, the time for one cycle), frequency (f = 1/T, in hertz) and amplitude to describe waves and vibrations. Those words mean the same thing here.

What makes motion simple harmonic?

An object moves with simple harmonic motion when the net force on it is a restoring force proportional to its displacement from equilibrium:

F_net = −kx

For a mass on an ideal spring on a frictionless surface, Hooke's law gives exactly this. From Newton's second law, ma = −kx, so

a = −(k/m)x

The acceleration is always opposite to the displacement and proportional to it. That one property is what makes the motion SHM.

Key words:

  • Equilibrium position (x = 0): where the net force is zero.
  • Amplitude (A): the largest displacement from equilibrium. The mass moves between x = +A and x = −A.
  • Period (T): the time for one complete cycle (for example, +A to −A and back to +A). Frequency f = 1/T.
t = 0:v = 0; a max, to the leftat = T/4:v max, to the left; a = 0vt = T/2:v = 0; a max, to the rightat = 3T/4:v max, to the right; a = 0vx = 0−A+A
A mass on a spring through one cycle. At the ends (x = ±A) it stops for an instant and its acceleration is largest; at x = 0 it moves fastest and its acceleration is zero. The acceleration always points toward x = 0.

The period of a mass on a spring

Experiments and theory both give

T = 2π√(m/k)

(You will derive it in the next lesson, SHM, Energy and the Reference Circle.) Notice:

  • A larger mass gives a longer period: it is harder to speed up and slow down.
  • A stiffer spring (larger k) gives a shorter period: the restoring force is bigger.
  • The period does not depend on the amplitude. A big oscillation and a small one take the same time, because the bigger one moves faster.
  • A mass hanging from a vertical spring has the same period. Gravity just shifts the equilibrium position down to where kx₀ = mg; the mass then oscillates about that new point.
tx+A−Aone period Tturning points: v = 0, |a| largestcrossing x = 0: |v| largest, a = 0x = A cos(2πt/T)
Position against time for a mass released from x = +A. The graph is a cosine curve: the mass is momentarily at rest at each peak and trough and moves fastest as it crosses x = 0.

The graph is a sinusoid

If the mass is released from rest at x = +A when t = 0, its position is

x = A cos(2πt/T)

with the angle in radians (as in the MHF4U unit Radian Measure and Trigonometric Functions). The velocity graph is a sine curve that is zero at the turning points, and the acceleration graph is the position graph flipped upside down (a = −(k/m)x).

Worked example

Example 1: Period and frequency

A 0.50 kg block oscillates on a horizontal spring with k = 200 N/m. Find its period and frequency.

  1. T = 2π√(m/k) = 2π√(0.50 ÷ 200) = 2π√0.0025 = 2π(0.050).
  2. T = 0.31 s (0.314 s).
  3. f = 1/T = 3.2 Hz.

T ≈ 0.31 s; f ≈ 3.2 Hz (about three bounces per second).

Worked example

Example 2: Finding k from a timing

A 0.25 kg mass on a spring makes 10 complete oscillations in 6.0 s. Find k.

  1. T = 6.0 s ÷ 10 = 0.60 s. (Timing 10 cycles and dividing reduces reaction-time error.)
  2. Square T = 2π√(m/k): T² = 4π²m/k, so k = 4π²m/T².
  3. k = 4π²(0.25) ÷ (0.60)² = 9.870 ÷ 0.36 = 27 N/m (27.4 N/m).

k ≈ 27 N/m.

Worked example

Example 3: Acceleration at a point

The block in Example 1 oscillates with amplitude 5.0 cm. Find its acceleration (a) at x = +4.0 cm, (b) at the turning points, (c) at equilibrium.

  1. k/m = 200 ÷ 0.50 = 400 s⁻².
  2. (a) a = −(400)(0.040) = −16 m/s² (toward equilibrium).
  3. (b) At x = ±0.050 m: |a| = (400)(0.050) = 20 m/s², the maximum, pointing toward x = 0.
  4. (c) At x = 0: a = 0.

−16 m/s²; 20 m/s² (max); 0. That maximum is about twice g, even for a gentle-looking bounce.

Worked example

Example 4: A vertical spring

A mass hung gently on a vertical spring stretches it 9.8 cm and hangs at rest. It is then pulled down a little and released. Find the period.

  1. At equilibrium kx₀ = mg, so k/m = g/x₀.
  2. T = 2π√(m/k) = 2π√(x₀/g) = 2π√(0.098 ÷ 9.8) = 2π√0.010 = 2π(0.10).
  3. T = 0.63 s. The mass and k are not needed separately.

About 0.63 s.

Just for fun

A mass on a spring is the most dedicated commuter in physics: it travels back and forth all day, never gets lost, and always takes exactly the same time for the trip, no matter how far it goes.

Hands-on activity

Time a bouncing spring

You need: a long rubber band or a Slinky-type spring (or a few rubber bands looped together), a plastic grocery bag, 4 identical cans of food, a kitchen scale, a phone stopwatch, and a sturdy hook or cupboard handle. Safety: keep your face away from the stretched band.

  1. Hang one can in the bag. Pull it down 2 to 3 cm and release. Time 10 complete up-and-down oscillations and divide by 10 to get T.
  2. Repeat with 2, 3 and 4 cans. Also repeat one trial with a larger pull-down to see whether the amplitude changes T.
  3. Make a table of mass m and T². Plot T² against m.
  4. If the band acts like an ideal spring, the graph is a straight line with slope 4π²/k. Find k.

What to look for: T should increase with mass, but not in proportion: four cans should take roughly twice as long per oscillation as one can (T ∝ √m). The amplitude should have little effect. Rubber bands are not perfect springs, so expect a little scatter (the mass of the bag and band also adds slightly).

Your turn

1

Find the period of a 2.0 kg mass on a spring with k = 50 N/m.

2

An oscillator has a frequency of 2.0 Hz. What is its period?

3

The mass on a spring is made four times as large. What happens to the period?

4

Where in its motion does a mass in SHM have its greatest speed? Its greatest acceleration?

5

A 0.40 kg mass on a spring has a period of 0.80 s. Find k.

6

The amplitude of an ideal mass-spring oscillator is doubled. What happens to the period?

7

A 0.30 kg mass is on a spring with k = 120 N/m. Find its acceleration when x = +3.0 cm.

More practice: warm-up

1

Define simple harmonic motion.

2

Write the formula for the period of a mass on a spring.

3

What is the relationship between period and frequency?

4

What is the amplitude of an oscillation?

5

At the turning points of SHM, what is the speed?

More practice: on your own

1

A mass on a spring makes 15 oscillations in 9.0 s. Find its period and frequency.

2

What mass gives a period of 1.0 s on a spring with k = 100 N/m?

3

A mass hangs at rest on a vertical spring, stretching it 4.0 cm. Find the period of small oscillations.

4

A 0.20 kg mass on a spring (k = 80 N/m) oscillates with amplitude 5.0 cm. Find its maximum acceleration.

5

A 1200 kg car rests on springs with a combined k of 1.2 × 10⁵ N/m. With worn-out shock absorbers it bounces freely after a bump. Find the period and frequency of the bounce.

Challenge

1

A mass oscillates on two identical springs. By what factor is the period with the springs in series larger than with the springs in parallel?

2

Astronauts on the International Space Station cannot use bathroom scales, so their mass is measured with a spring-driven device that oscillates. Suppose an empty seat on a spring with k = 600 N/m oscillates with T = 1.0 s, and with an astronaut aboard T = 2.4 s. Find the astronaut's mass.

3

Show that √(m/k) has units of seconds.

Check your understanding

Quiz

1.

Which condition defines simple harmonic motion?

2.

A 0.80 kg mass oscillates on a spring with k = 320 N/m. Its period is closest to:

3.

At the equilibrium position, a mass in SHM has:

4.

To double the period of a mass-spring oscillator, you could:

5.

A mass hanging on a vertical spring is compared with the same mass and spring lying horizontally on a frictionless table. The periods are:

6.

An oscillator completes 20 cycles in 5.0 s. Its frequency is:

For parents and teachers

Big idea (C3.2, C3.1): when a restoring force is proportional to displacement (Hooke's law), an object oscillates in simple harmonic motion with period T = 2π√(m/k), whatever the amplitude.

How to help: do the bouncing-spring activity together and time 10 oscillations with a phone. Ask your teen to predict what happens to the period when the mass is quadrupled, then test it.

Common difficulties:

  • Thinking the period depends on the amplitude.
  • Mixing up where speed and acceleration are largest (speed at the middle, acceleration at the ends).
  • Forgetting the square root: four times the mass doubles the period.
  • Timing one oscillation instead of 10 (reaction time then causes big errors).