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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

Cars have shock absorbers to damp the oscillations that would otherwise occur when the springs that attach the wheels to the car's frame are compressed or stretched. Ideally, the shock absorbers provide critical damping. If the shock absorbers fail, they provide less damping, resulting in an underdamped motion. You can perform a simple test of your shock absorbers by pushing down on one corner of your car and then quickly releasing it If this results in an up-and- down oscillation of the car, you know that your shock absorbers need changing. The spring on each wheel of a car has a spring constant of \(4005 \mathrm{~N} / \mathrm{m}\), and the car has a mass of \(851 \mathrm{~kg}\), equally distributed over all four wheels. Its shock absorbers have gone bad and provide only \(60.7 \%\) of the damping they were initially designed to provide. What will the period of the underdamped oscillation of this car be if the pushing-down shock absorber test is performed?

A 3.0 -kg mass is vibrating on a spring. It has a resonant angular speed of \(2.4 \mathrm{rad} / \mathrm{s}\) and a damping angular speed of \(0.14 \mathrm{rad} / \mathrm{s}\). If the driving force is \(2.0 \mathrm{~N}\), find the maximum amplitude if the driving angular speed is (a) \(1.2 \mathrm{rad} / \mathrm{s},\) (b) \(2.4 \mathrm{rad} / \mathrm{s}\), and \((\) c) \(4.8 \mathrm{rad} / \mathrm{s}\).

A cylindrical can of diameter \(10.0 \mathrm{~cm}\) contains some ballast so that it floats vertically in water. The mass of can and ballast is \(800.0 \mathrm{~g}\), and the density of water is \(1.00 \mathrm{~g} / \mathrm{cm}^{3}\) The can is lifted \(1.00 \mathrm{~cm}\) from its equilibrium position and released at \(t=0 .\) Find its vertical displacement from equilibrium as a function of time. Determine the period of the motion. Ignore the damping effect due to the viscosity of the water.

In a lab, a student measures the unstretched length of a spring as \(11.2 \mathrm{~cm}\). When a 100.0 - g mass is hung from the spring, its length is \(20.7 \mathrm{~cm}\). The mass-spring system is set into oscillatory motion, and the student obscrves that the amplitude of the oscillation decreases by about a factor of 2 after five complete cycles. a) Calculate the period of oscillation for this system, assuming no damping. b) If the student can measure the period to the nearest \(0.05 \mathrm{~s}\). will she be able to detect the difference between the period with no damping and the period with damping?

Two springs, each with \(k=125 \mathrm{~N} / \mathrm{m}\), are hung vertically, and \(1.00-\mathrm{kg}\) masses are attached to their ends. One spring is pulled down \(5.00 \mathrm{~cm}\) and released at \(t=0\); the othen is pulled down \(4.00 \mathrm{~cm}\) and released at \(t=0.300 \mathrm{~s}\). Find the phase difference, in degrees, between the oscillations of the two masses and the equations for the vertical displacements of the masses, taking upward to be the positive direction.
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