Astronauts on the first trip to Mars take along a pendulum that has a period on earth of 1.50 s. The period on Mars turns out to be 2.45 s. What is the free-fall acceleration on Mars?

Verified step by step guidance

1. Step 1: Recall the formula for the period of a pendulum: $T\; =\; 2\pi \surd (L/g)$, where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.
2. Step 2: Rearrange the formula to solve for g: $g\; =\; (4\pi ²L)/T²$. This equation shows that g is inversely proportional to the square of the period.
3. Step 3: Since the pendulum's length L remains constant, use the ratio of the periods on Earth and Mars to find the ratio of the gravitational accelerations: $g\_\{Mars\}/g\_\{Earth\}\; =\; (T\_\{Earth\}/T\_\{Mars\})²$.
4. Step 4: Substitute the given values for the periods: $T\_\{Earth\}\; =\; 1.50\; \backslash ,\; s$ and $T\_\{Mars\}\; =\; 2.45\; \backslash ,\; s$. Calculate the ratio $(T\_\{Earth\}/T\_\{Mars\})²$.
5. Step 5: Multiply the known value of Earth's gravitational acceleration, $g\_\{Earth\}\; =\; 9.8\; \backslash ,\; m/s²$, by the calculated ratio to find $g\_\{Mars\}$.

Key Concepts

Pendulum Period

The period of a pendulum is the time it takes to complete one full oscillation. It is influenced by the length of the pendulum and the acceleration due to gravity. The formula for the period (T) of a simple pendulum is T = 2π√(L/g), where L is the length and g is the acceleration due to gravity. Understanding this relationship is crucial for analyzing how the period changes in different gravitational fields.

Acceleration due to Gravity

Acceleration due to gravity (g) is the rate at which an object accelerates towards the center of a celestial body, such as Earth or Mars. On Earth, g is approximately 9.81 m/s², while on Mars, it is significantly lower, around 3.71 m/s². The difference in g affects the motion of objects, including pendulums, and is essential for calculating the gravitational force experienced by astronauts on Mars.

Gravitational Comparison

When comparing gravitational effects on different planets, the change in the period of a pendulum can be used to derive the gravitational acceleration. By knowing the periods on Earth and Mars, one can set up a ratio based on the pendulum period formula to solve for the unknown gravitational acceleration on Mars. This concept illustrates how physical principles can be applied across different environments in space.