γ°οΈPendulum Motion
The Simple Pendulum
A simple pendulum consists of a mass (the "bob") on a string of length L, swinging from a fixed point.
For small angles (< 15Β°), the restoring force is approximately proportional to displacement, making it SHM. The period is:
T = 2Οβ(L/g)
Remarkable features: - Period depends on LENGTH only β not mass, not amplitude (for small angles)! - Longer pendulum β longer period - On the Moon (smaller g) β longer period (pendulums swing slower)
This is why Galileo could use his pulse to time pendulum swings β both heart rate and pendulum period were reproducible!
Think About It
A grandfather clock pendulum has a 2-second period (1 second each way). If you take this clock to the top of a high mountain where g is slightly smaller, does the clock run fast or slow?
βοΈ Worked Example
Problem: A pendulum has a 1.5 m string. Find its period. (g = 9.8 m/sΒ²)
π Key Equations
Pendulum
T_pendulum = 2Οβ((L)/(g))f = (1)/(2Ο)β((g)/(L))ΞΈ < 15Β° for SHM approximationβ οΈ Common Mistakes
Misconception: Thinking that a heavier pendulum bob swings with a shorter period.
β Correct thinking: Mass cancels out of the pendulum period equation: T = 2Οβ(L/g). A 1 kg bob and a 100 kg bob on the same length string have identical periods.
Why: Gravity pulls harder on a heavier bob, but the heavier bob also requires more force to accelerate. These effects cancel exactly, just like in free fall.
Misconception: Using T = 2Οβ(L/g) for large-angle swings (e.g., 60Β° or 90Β°).
β Correct thinking: This formula is only valid for small angles (< ~15Β°). For large angles, the restoring force is no longer proportional to displacement, so the motion isn't true SHM and the period is longer.
Why: The SHM approximation uses sin ΞΈ β ΞΈ (in radians), which only holds for small angles. At 15Β°, the error is about 0.5%; at 30Β° it's about 2%.
Misconception: Confusing the effect of g on the pendulum period β thinking lower g means faster swings.
β Correct thinking: Lower g means a LONGER period. T = 2Οβ(L/g): if g decreases, L/g increases, making T larger. On the Moon, pendulums swing more slowly.
Why: Less gravitational pull means the restoring force is weaker, so the pendulum accelerates less and takes longer to complete each swing.
π Practice Problems
Try these problems. Check your answer when ready.
A pendulum has a length of 0.25 m. Find its period. (g = 9.8 m/sΒ²)
A pendulum has a period of 3 s on Earth (g = 9.8 m/sΒ²). What is its length?
A grandfather clock uses a pendulum with T = 2 s. If you move this clock to the Moon (g_Moon = 1.62 m/sΒ²), what will its new period be?
Pendulum A has length L and period T. Pendulum B has length 4L. What is the period of pendulum B?
T = 2Οβ((L)/(g))A geologist measures a pendulum's period to be 2.01 s in a remote location (pendulum length = 0.993 m). What is the local value of g?
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