Question

Imagine you are an astronaut who has landed on another planet and wants to determine the free-fall acceleration on that planet. In one of the experiments you decide to conduct, you use a pendulum \(0.500 \mathrm{~m}\) long and find that the period of oscillation for this pendulum is \(1.50 \mathrm{~s}\). What is the acceleration due to gravity on that planet?

Short answer

Answer: The acceleration due to gravity on that planet is approximately \(9.60 \mathrm{~m/s^2}\).

Step-by-step solution

Write down the formula for the period of a pendulum

The formula for the period of a pendulum is given by: \(T = 2\pi \sqrt{\frac{L}{g}}\) Where \(T\) is the period of oscillation, \(L\) is the length of the pendulum, and \(g\) is the acceleration due to gravity. We are given the values of \(T\) and \(L\), and we need to find the value of \(g\).

Plug in the given values

We are given that the period of oscillation for this pendulum is \(1.50 \mathrm{~s}\) and the length of the pendulum is \(0.500 \mathrm{~m}\). Plugging these values into the formula, we get: \(1.50 = 2\pi \sqrt{\frac{0.500}{g}}\)

Solve for \(g\)

Now we need to solve the equation for \(g\). First, we can isolate the square root term by dividing both sides of the equation by \(2\pi\): \(\frac{1.50}{2\pi} = \sqrt{\frac{0.500}{g}}\) Next, square both sides of the equation to eliminate the square root: \(\left(\frac{1.50}{2\pi}\right)^2 = \frac{0.500}{g}\) Now we can solve for \(g\) by multiplying both sides by \(g\) and then dividing by \(\left(\frac{1.50}{2\pi}\right)^2\): \(g = \frac{0.500}{\left(\frac{1.50}{2\pi}\right)^2}\)

Calculate the value of \(g\)

Now we can plug in the numbers and calculate the value of \(g\): \(g = \frac{0.500}{\left(\frac{1.50}{2\pi}\right)^2} \approx 9.60 \mathrm{~m/s^2}\) So, the acceleration due to gravity on that planet is approximately \(9.60 \mathrm{~m/s^2}\).

Key concepts

Pendulum Period Formula

When exploring the timely swings of a pendulum, we come across an important concept known as the pendulum period formula. It is an expression that allows us to determine the time it takes for a pendulum to complete one full oscillation, also known as its period. The formula is written as:

\[\begin{equation}T = 2\textbackslashpi \textbackslashsqrt{\frac{L}{g}}\textbackslashend{equation}
Here, \(T\) represents the period of the pendulum, \(L\) is the length of the pendulum's string, and \(g\) stands for the acceleration due to gravity. By rearranging this formula, it becomes possible to solve for any of the three variables if the other two are known. For instance, when trying to measure the gravitational acceleration on an unknown planet, as in our astronaut's experiment, the pendulum period formula becomes a crucial tool for analysis. It's a demonstration of how classical mechanics can be used in practical and creative ways to examine the characteristics of celestial bodies.

Oscillation

The concept of oscillation refers to the repetitive movement of an object between two points. In the context of pendulums, oscillation describes the motion of the pendulum swinging back and forth from its highest point on one side to its highest point on the other side.Understanding oscillation is fundamental when studying various forms of periodic motions. A single oscillation encompasses one complete cycle, which includes the pendulum swinging to the left, swinging to the right, and returning to its original position. By monitoring the time it takes to complete this motion - the period - we can analyze the pendulum's oscillatory pattern. Through careful observation and measurement, oscillations can also reveal information about the properties of the system in which they occur, such as the gravitational pull in different locations.

Free-fall Acceleration

Finally, the term free-fall acceleration is used to describe the acceleration of an object solely under the influence of gravity. This value is often denoted by the symbol \(g\), and on Earth, it is approximately \(9.81\, \text{m/s}^2\). However, this value can vary slightly depending on altitude and geographical location.

In the case of our adventurous astronaut, who is conducting experiments on an alien planet, determining this planetary constant is key. By using the pendulum period formula and the concept of oscillation, the astronaut can calculate the free-fall acceleration specific to the planet. This knowledge is invaluable for many scientific and practical purposes, including the planning of future space missions and the understanding of how other physical processes might differ on that planet compared to Earth. It's fascinating how the simple motion of a pendulum can unlock answers to profound questions about the cosmos.