A certain simple pendulum has a period on the earth of 1.60 s. What is its period on the surface of Mars, where g = 3.71 m/s^2?

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. Calculate the length of the pendulum using the given period on Earth and the acceleration due to gravity on Earth ($g_{earth} = 9.81 \text{ m/s}^2$). Rearrange the formula to solve for $L$: $L = \frac{g_{earth} T_{earth}^2}{4\pi^2}$.
3. Substitute the known values for $T_{earth}$ and $g_{earth}$ into the rearranged formula to find the length $L$ of the pendulum.
4. Use the length $L$ calculated from the Earth data and the given acceleration due to gravity on Mars ($g_{mars} = 3.71 \text{ m/s}^2$) to find the period of the pendulum on Mars. Substitute these values into the original period formula: $T_{mars} = 2\pi \sqrt{\frac{L}{g_{mars}}}$.
5. Simplify the expression to find the theoretical period of the pendulum on Mars, which will show how the period changes with a different gravitational acceleration.

Key Concepts

Simple Pendulum

A simple pendulum consists of a mass (the bob) attached to a string or rod of negligible mass, swinging 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.

Period of a Pendulum

The period of a pendulum is given by the formula T = 2π√(L/g), where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity. This relationship shows that the period is directly influenced by the gravitational force acting on the pendulum. A lower gravitational acceleration results in a longer period.

Acceleration due to Gravity (g)

Acceleration due to gravity is the rate at which an object accelerates towards the Earth (or another celestial body) due to gravitational force. On Earth, g is approximately 9.81 m/s², while on Mars, it is about 3.71 m/s². This difference in g affects the behavior of pendulums and other objects in motion, leading to variations in their periods and other dynamic properties.