A simple pendulum is a mass called a bob hanging from a light string that swings back and forth under gravity. It is one of the clearest examples of periodic motion, where a system repeats its motion in a regular cycle. Pendulums matter because they connect forces, energy, and oscillations in a system that is easy to visualize.
They also show how a complicated curved motion can become simple when the swing angle is small.
For small angles, the pendulum behaves almost like simple harmonic motion because the restoring force is nearly proportional to the displacement. The component of gravity along the arc pulls the bob back toward equilibrium, while the string tension mainly keeps the bob moving in a circular path. The period depends mostly on the length of the string and the strength of gravity, not on the mass of the bob.
This is why the formula T = 2π sqrt(L/g) is so useful for predicting pendulum motion.
Key Facts
- Period for a small-angle simple pendulum: T = 2π sqrt(L/g).
- Frequency is the inverse of period: f = 1/T.
- For small angles in radians, sin θ ≈ θ.
- Restoring force along the arc: F_restore = -mg sin θ.
- Small-angle approximation gives F_restore ≈ -mgθ.
- For small angles, the period does not depend on the bob's mass or the swing amplitude.
Vocabulary
- Simple pendulum
- A simple pendulum is an idealized system made of a point mass suspended from a massless string that swings under gravity.
- Period
- The period is the time required for one complete back-and-forth cycle of motion.
- Equilibrium position
- The equilibrium position is the lowest vertical position where the pendulum would hang at rest.
- Restoring force
- The restoring force is the force component that pulls the bob back toward the equilibrium position.
- Small-angle approximation
- The small-angle approximation is the rule that sin θ is nearly equal to θ when θ is measured in radians and is small.
Common Mistakes to Avoid
- Using degrees in sin θ ≈ θ, which is wrong because the approximation only works when θ is measured in radians.
- Including the bob's mass in the period formula, which is wrong because mass cancels out for an ideal small-angle pendulum.
- Assuming the tension is the restoring force, which is wrong because the restoring force is the component of gravity along the arc.
- Applying T = 2π sqrt(L/g) to very large angles without correction, which is wrong because the small-angle approximation becomes inaccurate.
Practice Questions
- 1 A pendulum has length L = 1.20 m on Earth where g = 9.8 m/s^2. Calculate its period using T = 2π sqrt(L/g).
- 2 A clock pendulum must have a period of 2.00 s on Earth. What length L is needed if g = 9.8 m/s^2?
- 3 Two pendulums have the same length and swing through small angles, but one bob has twice the mass of the other. Explain whether their periods are the same or different, and why.