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Chapter 10: Problem 62

A pendulum clock has a period of 0.650 s on Earth. It is taken to another planet and found to have a period of \(0.862 \mathrm{s}\). The change in the pendulum's length is negligible. (a) Is the gravitational field strength on the other planet greater than or less than that on Earth? (b) Find the gravitational field strength on the other planet.

Short Answer

Expert verified
Answer: The gravitational field strength on the other planet is approximately 5.57 m/s².

Step by step solution

01

Find the ratio of the periods

Divide the period of the pendulum on the other planet \(T_2\) by its period on Earth \(T_1\). $$\frac{T_2}{T_1} = \frac{0.862 \mathrm{s}}{0.650 \mathrm{s}}$$ Calculate the value of the ratio: $$\frac{T_2}{T_1} = 1.326$$
02

Square the ratio of the periods

Square the ratio \(\frac{T_2}{T_1}\) to obtain the ratio of \(\frac{l_2}{g_2}\) to \(\frac{l_1}{g_1}\). $$\left(\frac{T_2}{T_1}\right)^2 = \left(\frac{1.326}{1}\right)^2 = 1.76$$
03

Determine the relationship between the gravitational field strengths

Since the lengths of the pendulum are negligibly different (the same), we can write the following equation for the ratio of gravitational field strengths: $$\frac{l_2}{g_2} = \frac{l_1}{g_1}$$ Since the ratio obtained in Step 2 is greater than 1, we conclude that the gravitational field strength on the other planet is less than that on Earth: $$(l_1 = l_2) \Rightarrow g_2 < g_1$$
04

Find the gravitational field strength on the other planet

As we know the ratio of \(g_1\) to \(g_2\) after obtaining the ratio of periods in Step 2, we can find the value of \(g_2\) using Earth's gravitational field strength, \(g_1 = 9.81 \mathrm{m/s}^2\). $$g_2 = \frac{g_1}{1.76}$$ $$g_2 = \frac{9.81 \mathrm{m/s^2}}{1.76}$$ Calculate the gravitational field strength on the other planet: $$g_2 = 5.57 \mathrm{m/s^2}$$ So the gravitational field strength on the other planet is approximately 5.57 m/s².

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Most popular questions from this chapter

In an aviation test lab, pilots are subjected to vertical oscillations on a shaking rig to see how well they can recognize objects in times of severe airplane vibration. The frequency can be varied from 0.02 to $40.0 \mathrm{Hz}\( and the amplitude can be set as high as \)2 \mathrm{m}$ for low frequencies. What are the maximum velocity and acceleration to which the pilot is subjected if the frequency is set at \(25.0 \mathrm{Hz}\) and the amplitude at \(1.00 \mathrm{mm} ?\)
An object of mass \(306 \mathrm{g}\) is attached to the base of a spring, with spring constant \(25 \mathrm{N} / \mathrm{m},\) that is hanging from the ceiling. A pen is attached to the back of the object, so that it can write on a paper placed behind the mass-spring system. Ignore friction. (a) Describe the pattern traced on the paper if the object is held at the point where the spring is relaxed and then released at \(t=0 .\) (b) The experiment is repeated, but now the paper moves to the left at constant speed as the pen writes on it. Sketch the pattern traced on the paper. Imagine that the paper is long enough that it doesn't run out for several oscillations.
The air pressure variations in a sound wave cause the eardrum to vibrate. (a) For a given vibration amplitude, are the maximum velocity and acceleration of the eardrum greatest for high-frequency sounds or lowfrequency sounds? (b) Find the maximum velocity and acceleration of the eardrum for vibrations of amplitude \(1.0 \times 10^{-8} \mathrm{m}\) at a frequency of $20.0 \mathrm{Hz}\(. (c) Repeat (b) for the same amplitude but a frequency of \)20.0 \mathrm{kHz}$.
A bungee jumper leaps from a bridge and undergoes a series of oscillations. Assume \(g=9.78 \mathrm{m} / \mathrm{s}^{2} .\) (a) If a \(60.0-\mathrm{kg}\) jumper uses a bungee cord that has an unstretched length of \(33.0 \mathrm{m}\) and she jumps from a height of \(50.0 \mathrm{m}\) above a river, coming to rest just a few centimeters above the water surface on the first downward descent, what is the period of the oscillations? Assume the bungee cord follows Hooke's law. (b) The next jumper in line has a mass of \(80.0 \mathrm{kg} .\) Should he jump using the same cord? Explain.
A pendulum is made from a uniform rod of mass \(m_{1}\) and a small block of mass \(m_{2}\) attached at the lower end. (a) If the length of the pendulum is \(L\) and the oscillations are small, find the period of the oscillations in terms of \(m_{1}, m_{2}, L,\) and \(g .\) (b) Check your answer to part (a) in the two special cases \(m_{1} \gg m_{2}\) and \(m_{1}<<m_{2}\).
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