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Chapter 14: Problem 63

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 \(0.50 \mathrm{~m}\) 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

Expert verified
Answer: The acceleration due to gravity on this planet is approximately \(6.076 \mathrm{~m/s^2}\).

Step by step solution

01

Write down the known values

We know the length of the pendulum, L = 0.50 m and the period of oscillation, T = 1.50 s.
02

Write the formula for the period of oscillation

The formula for the period of oscillation of a pendulum is given by \(T=2\pi\sqrt{\frac{L}{g}}\).
03

Plug the known values into the formula

We know the values of T and L, so our equation becomes: \(1.50 = 2\pi\sqrt{\frac{0.50}{g}}\)
04

Solve for the acceleration due to gravity, g

To solve for g, we need to isolate g in the equation. First, we can divide both sides of the equation by 2\(\pi\), which results in: \(\frac{1.50}{2\pi}=\sqrt{\frac{0.50}{g}}\) Next, we square both sides of the equation to eliminate the square root: \(\left(\frac{1.50}{2\pi}\right)^2 = \frac{0.50}{g}\) Now, we can cross multiply to isolate g: \(g = \frac{0.50}{\left(\frac{1.50}{2\pi}\right)^2}\)
05

Compute the value of g

Compute the value of g using a calculator: \(g = \frac{0.50}{\left(\frac{1.50}{2\pi}\right)^2} \approx 6.076 \mathrm{~m/s^2}\)
06

State the final answer

The acceleration due to gravity on this planet is approximately \(6.076 \mathrm{~m/s^2}\).

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Most popular questions from this chapter

The relative motion of two atoms in a molecule can be described as the motion of a single body of mass \(m\) moving in one dimension, with potcntial encrgy \(U(r)=A / r^{12}-B / r^{6}\) where \(r\) is the separation between the atoms and \(A\) and \(B\) are positive constants. a) Find the equilibrium separation, \(r_{0}\) of the atoms, in terms of the constants \(A\) and \(B\) b) If moved slightly, the atoms will oscillate about their equilibrium separation. Find the angular frequency of this oscillation, in terms of \(A, B,\) and \(m\).

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