Simple Harmonic Motion - Physics 101 / AP Physics 1 Review with Dianna Cowern

Main Ideas

What Simple Harmonic Motion Is

Simple harmonic motion (SHM) is back-and-forth motion caused by a restoring force that points toward equilibrium and is proportional to displacement:

[ F = -Cx ]

The minus sign indicates that the force acts opposite to the displacement. This linear restoring force produces sinusoidal motion: position varies like a sine or cosine, with one main frequency.

SHM differs from oscillatory or harmonic motion more broadly. A bouncing ball and the balls in a Newton’s cradle move back and forth, for example, but their motion is not a single, smooth sine wave.

Pendulums and the Small-Angle Approximation

A pendulum trades gravitational potential energy for kinetic energy. It has the most potential energy at the ends of its swing and moves fastest at the bottom. In an idealized system with no air resistance, it returns to the same height and repeats with a consistent period.

The period is the time required for one complete cycle. In the workshop demonstration, the bowling ball’s measured period was about 4.125 seconds.

For a pendulum at a small angle, (\sin\theta \approx \theta), with the angle measured in radians. The restoring force is therefore approximately proportional to displacement:

[ F \approx -\frac{mg}{L}x ]

Here, (m) is the pendulum’s mass, (g) is gravitational acceleration, and (L) is the string length. Combining this relationship with Newton’s second law shows that the pendulum’s mass cancels. Its period depends on its length and the local gravitational acceleration—not on the bob’s mass.

Pendulum Period and the Tree-Height Estimate

Using the sinusoidal-motion model gives the pendulum’s angular frequency, (\omega = \sqrt{g/L}), and its period:

[ T = 2\pi\sqrt{\frac{L}{g}} ]

The lesson uses this formula to estimate the height of the tree branch:

1. Approximate the measured period as (4) seconds and take (g \approx 10\ \text{m/s}^2).
2. Solve for the pendulum length, giving about (4) metres.
3. Add about (1) metre because the ball hangs roughly that far above the ground at rest.
4. Estimate the branch height as about 5 metres.

A one-metre pendulum has a period close to two seconds, or about one second per half-swing. The video connects this to historical attempts to use a seconds-based pendulum as a standard for measuring a metre. It also notes that pendulum clocks must keep their length stable as temperature changes; Big Ben’s pendulum uses materials arranged to counteract expansion and contraction.

Springs and Their Period

A spring follows Hooke’s law, (F = -kx). Newton’s second law then gives an acceleration proportional to displacement, with the mass affecting how quickly the system oscillates.

The period of an ideal mass-spring oscillator is:

[ T = 2\pi\sqrt{\frac{m}{k}} ]

A larger mass makes the period longer, while a stiffer spring (larger (k)) makes it shorter.

For the car example, the lesson assumes a (1{,}000)-kg car oscillating at (1) hertz, so its period is (1) second. The total spring constant is about 40,000 N/m. With four identical springs in parallel, that is about 10,000 N/m per spring.

A spring can provide a restoring force for a clock in space, where a pendulum cannot rely on gravity in the same way.

Position, Velocity, and Acceleration in SHM

In the pendulum model, position varies sinusoidally, velocity is also sinusoidal but shifted in phase, and the restoring acceleration points opposite the displacement.

At the ends of the swing, displacement is greatest and speed is momentarily zero. At the equilibrium point, displacement is zero and speed is greatest.

Caveat: The lesson’s statement that there is “no acceleration” at the pendulum’s lowest point refers to acceleration along the oscillation’s path—the restoring or tangential acceleration. A real pendulum still has centripetal acceleration there because its direction of motion is changing.

Examples and Limits

Examples presented as approximately simple harmonic include:

• A small-angle pendulum or swing
• A mass on a spring
• A vibrating guitar string
• A marble rolling in a bowl
• A vibrating doorstop or tuning fork
• A bobbing duck
• Air vibrating in a bottle
• Car suspension

A bouncing ball and a Newton’s-cradle ball are given as examples that are not simple harmonic because their motion does not follow one smooth sinusoidal pattern.

The pendulum result is an approximation that works for small swings. SHM remains useful because many real oscillations can be treated as approximately sinusoidal, even when additional effects are present.

Speakers and Sources Featured

• Dianna Cowern — Host and instructor.
• Kyle — Special guest who describes applying physics in the U.S. Space Force, including orbital motion, satellite communications, rocket operations, and orbital mechanics.
• Simone — Mentioned as the friend who hosted the bowling-ball demonstration; she does not speak in the subtitles.