You pull a simple pendulum 0.240 m long to the side through an angle of 3.50° and release it. How much time does it take the pendulum bob to reach its highest speed?

Verified step by step guidance

1. Understand that the pendulum's motion can be approximated as simple harmonic motion for small angles. The angle given, 3.50°, is small enough to use this approximation.
2. The period of a simple pendulum is given by the formula: $T=2\pi \sqrt{\frac{l}{g}}$, where $l$ is the length of the pendulum and $g$ is the acceleration due to gravity (approximately 9.81 m/s²).
3. Calculate the period $T$ using the given length of the pendulum, $l=0.240 m$.
4. The pendulum reaches its highest speed at the lowest point of its swing, which is halfway through its period. Therefore, the time to reach the highest speed is half of the period: $\frac{T}{2}$.
5. Substitute the calculated period into the expression for half the period to find the time it takes for the pendulum bob to reach its highest speed.

Key Concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) describes the oscillatory motion of systems like pendulums, where the restoring force is proportional to the displacement. In a simple pendulum, the motion is approximately SHM for small angles, allowing us to use formulas for period and frequency to analyze its motion.

Pendulum Period

The period of a simple pendulum, which is the time for one complete oscillation, depends on its length and the acceleration due to gravity. It is given by the formula T = 2π√(L/g), where L is the pendulum length and g is the gravitational acceleration. This formula helps determine the timing of various points in the pendulum's swing.

Kinetic and Potential Energy in Pendulums

In a pendulum, energy oscillates between kinetic and potential forms. At the highest point, energy is all potential, while at the lowest point, it is all kinetic. The pendulum reaches its highest speed at the lowest point, where potential energy is minimal and kinetic energy is maximal.