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 if the pendulum is released at an angle of 1.75° instead of 3.50°?

Verified step by step guidance

1. Understand that the period of a simple pendulum is given by the formula: $T=2\pi \surd \frac{l}{g}$, where $l$ is the length of the pendulum and $g$ is the acceleration due to gravity.
2. Note that the period is independent of the amplitude (angle) for small angles, which is a characteristic of simple harmonic motion. Therefore, the angle change from 3.50° to 1.75° does not affect the period.
3. Calculate the period using the given length of the pendulum, $l=0.240m$, and the standard value of gravitational acceleration, $g=9.81\mathrm{m/s²}$.
4. Substitute the values into the formula: $T=2\pi \surd \frac{0.240}{9.81}$.
5. Simplify the expression to find the period $T$, which is the time it takes for one complete oscillation of the pendulum.

Key Concepts

Simple Pendulum

A simple pendulum consists of a mass (called the bob) attached to a string or rod of fixed length, which swings back and forth under the influence of gravity. The motion is periodic and can be approximated as simple harmonic motion for small angles, where the restoring force is proportional to the displacement angle.

Period of a Pendulum

The period of a pendulum is the time it takes to complete one full cycle of its motion. For small angles, the period is approximately independent of the amplitude and is given by the formula T = 2π√(L/g), where L is the length of the pendulum and g is the acceleration due to gravity. This approximation holds true for angles less than about 15°.

Small Angle Approximation

The small angle approximation is used in pendulum motion to simplify calculations by assuming that sin(θ) ≈ θ when θ is measured in radians and is small. This allows the pendulum's motion to be treated as simple harmonic, making the period formula applicable. This approximation is valid for angles typically less than 15°, ensuring the pendulum behaves predictably.