A simple pendulum is a weight (or bob) suspended from a fixed point by a lightweight, inextensible string or rod. It is a classic example of a system that exhibits simple harmonic motion when displaced from its equilibrium position and released.
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The period of a simple pendulum's oscillation is determined by the length of the string and the acceleration due to gravity, and is independent of the mass of the bob.
The period of a simple pendulum is given by the formula: $T = 2\pi\sqrt{\frac{l}{g}}$, where $T$ is the period, $l$ is the length of the string, and $g$ is the acceleration due to gravity.
The frequency of a simple pendulum's oscillation is the inverse of its period, and is given by the formula: $f = \frac{1}{T}$.
The motion of a simple pendulum is an example of simple harmonic motion, where the restoring force is proportional to the displacement from the equilibrium position.
The amplitude of a simple pendulum's oscillation is the maximum displacement from the equilibrium position, and the energy of the system is divided between potential energy (at the extremes of the motion) and kinetic energy (at the midpoint of the motion).
Review Questions
Explain how the period of a simple pendulum's oscillation is related to the length of the string and the acceleration due to gravity.
The period of a simple pendulum's oscillation is determined by the length of the string and the acceleration due to gravity, as described by the formula $T = 2\pi\sqrt{\frac{l}{g}}$. This means that as the length of the string increases, the period of the oscillation also increases, and as the acceleration due to gravity increases, the period of the oscillation decreases. The period is independent of the mass of the bob, as long as the bob is much heavier than the string.
Describe the relationship between the period and frequency of a simple pendulum's oscillation.
The frequency of a simple pendulum's oscillation is the inverse of its period, as given by the formula $f = \frac{1}{T}$. This means that as the period of the oscillation increases, the frequency decreases, and vice versa. The frequency represents the number of oscillations or cycles completed by the pendulum in a given unit of time, while the period represents the time taken for one complete oscillation or cycle.
Analyze how the motion of a simple pendulum is an example of simple harmonic motion, and explain the relationship between the pendulum's displacement, restoring force, and energy.
The motion of a simple pendulum is an example of simple harmonic motion because the restoring force, which is the force that pulls the pendulum back towards its equilibrium position, is proportional to the displacement from that position. This means that the greater the displacement, the greater the restoring force, and the pendulum will oscillate back and forth around its equilibrium position. The energy of the system is divided between potential energy, which is greatest at the extremes of the motion where the displacement is maximum, and kinetic energy, which is greatest at the midpoint of the motion where the velocity is maximum. This cyclic transfer of energy between potential and kinetic forms is characteristic of simple harmonic motion.