You pull a simple pendulum 0.240 m long to the side through an angle of 3.50° and release it. (b) How much time does it take if the pendulum is released at an angle of 1.75° instead of 3.50°?

Verified step by step guidance

1. Identify the formula for the period of a simple pendulum, which is given by $T = 2\pi \sqrt{\frac{L}{g}}$, where $T$ is the period, $L$ is the length of the pendulum, and $g$ is the acceleration due to gravity.
2. Note that the period of the pendulum is independent of the amplitude (angle) as long as the angle is small. This is an approximation that holds true for small angular displacements (typically less than 15 degrees).
3. Since the length of the pendulum and the acceleration due to gravity remain constant, and the angle of release is small, the period $T$ remains the same regardless of whether the pendulum is released from 3.50° or 1.75°.
4. Calculate the period $T$ using the length $L = 0.240$ m. Substitute this value and the standard acceleration due to gravity (approximately $9.81 \text{ m/s}^2$) into the period formula.
5. The time for one complete oscillation (back and forth) is the period $T$. Since the period does not change with a change in the angle of release, the time taken for the pendulum released at 1.75° will be the same as calculated for 3.50°.

Key Concepts

Simple Pendulum

A simple pendulum consists of a mass (the bob) attached to a string of fixed length that swings back and forth under the influence of gravity. The motion is periodic, and the time it takes to complete one full cycle is called the period. The period depends on the length of the pendulum and the acceleration due to gravity, but for small angles, it is approximately independent of the amplitude.

Small Angle Approximation

The small angle approximation is a mathematical simplification used in pendulum motion, where angles less than about 15° can be approximated using the sine function. For small angles, sin(θ) is approximately equal to θ (in radians), which allows for simpler calculations of the pendulum's period. This approximation is crucial for determining the period when the pendulum is released at different angles.

Period of a Pendulum

The period of a pendulum is the time it takes to complete one full swing back and forth. For a simple pendulum, the period (T) can be calculated using the formula T = 2π√(L/g), where L is the length of the pendulum and g is the acceleration due to gravity. This formula shows that the period is influenced primarily by the length of the pendulum, and for small angles, it remains relatively constant even if the angle of release changes.