Simple Pendulum: Period, Length and Gravity

What can a swinging pendulum reveal?

A pendulum can be used to investigate periodic motion with surprisingly few variables. Release a small bob from a modest angle and it repeatedly moves through its lowest point. The period is the time required for one complete back-and-forth oscillation. It is not the time from one extreme to the opposite extreme; that is half a period.

Learning targets

Relate a pendulum’s period to string length and gravitational acceleration.
Explain why bob mass cancels in the ideal small-angle approximation.
Identify the difference between exact pendulum motion and approximate simple harmonic motion.
Use a simulation to measure period and interpret the exchange of kinetic and gravitational potential energy.

Forces and the restoring effect

For a small bob suspended by a light, inextensible string of length L, weight mg acts vertically downward and tension acts along the string. The tangential component of weight, −mg sin θ, pulls the bob toward equilibrium. Since arc displacement s = Lθ when θ is expressed in radians, for small angles sin θ ≈ θ ≈ s/L. The tangential equation becomes approximately maₜ = −(mg/L)s.

aₜ ≈ −(g/L)s

ω ≈ √(g/L)

T ≈ 2π√(L/g)

f = 1/T

This is the equation of simple harmonic motion for small angular displacement. The bob mass m cancels: a heavier ideal bob does not, by mass alone, oscillate more slowly.

Why the small-angle qualification matters

The exact restoring component contains sin θ rather than θ. At sufficiently large amplitude, the motion is still periodic but its period increases with amplitude. The familiar T formula is an approximation. In this site’s pendulum model the angle is kept small (3°–12°), a range suitable for demonstrating the approximation. Keep the angle fixed while testing length or gravity so that comparisons remain fair.

A worked period calculation

Take L = 1.00 m and g = 9.81 m/s². Then T ≈ 2π√(1/9.81) = 2.006 s. At L = 4.00 m and unchanged g, T ≈ 4.012 s. Increasing the length fourfold doubles the period rather than multiplying it by four. At fixed length, a planet with one-quarter the gravitational acceleration would also give twice the small-angle period.

Kinetic and potential energy

At the turning points the bob briefly has zero speed and maximum gravitational potential energy relative to its lowest position. At the lowest point speed and kinetic energy are greatest, while gravitational potential energy is smallest. In the ideal no-damping model, the total mechanical energy remains constant. In a real pendulum, friction and air drag gradually remove mechanical energy unless the motion is driven.

U(θ) = mgL(1 − cos θ) relative to the lowest point

K = ½mv²

Guided experiment: find the period law

Open the Simple Pendulum Lab. Select a small angle and leave it unchanged for the investigation.
Choose L = 1.0 m and Earth gravity. Predict the period; record the display reading or time several cycles.
Increase L where available. Plot or tabulate T² against L: the ideal relation is linear.
Keep L fixed and reduce g. Explain why a weaker gravitational restoring effect lengthens the period.
If a mass control is present, test whether changing bob mass changes the ideal period. Never assume a control exists if the simulation omits it.
Compare kinetic and potential energy at the middle and ends of the swing.

Common misconceptions

“A longer pendulum always moves faster.” Its period is longer; maximum linear speed depends on the amplitude and available potential-energy change.
“Mass controls the period.” It does not for an ideal small-angle pendulum.
“Period equals half a swing.” A complete oscillation requires a return to the initial position and direction.
“The small-angle equation remains exact at every angle.” It does not.

Practice with solutions

If length becomes 9L, what is the new period? Answer: 3T, because T ∝ √L.
If g becomes 4g at unchanged length, what happens to T? Answer: it halves.
A 1 m pendulum near Earth has a period close to 2.0 s. Does a 2 kg bob change this value compared with a 0.2 kg bob? Answer: not in the ideal small-angle approximation.
What is the period if 10 complete oscillations take 19.8 s? Answer: 19.8/10 = 1.98 s.

Curriculum and experimental thinking

The pendulum connects NCERT oscillations to gravitational acceleration and teaches experimental control. For a valid comparison, adjust only one variable at a time. Explain why measurements from a real classroom vary: angle-setting error, reaction time, pivot friction and the fact that a physical bob and string are not ideal mathematical objects.

Explore interactively

Use the free lab below, predict before changing settings, and compare your model with the worksheet. For a more advanced extension, investigate how the exact large-angle period differs from the small-angle prediction using an appropriate higher-fidelity model.

Frequently asked questions

What is one complete oscillation?

A return to the starting position with the same direction of motion.

Does a heavier bob swing more slowly?

No, not in the ideal small-angle model; mass cancels.

If length is multiplied by four, what happens to period?

It doubles, because T is proportional to the square root of length.

Is the period independent of amplitude at large angles?

No; the small-angle approximation becomes less accurate as amplitude grows.