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Chapter 14: Problem 1

Two children are on adjacent playground swings of the same height. They are given pushes by an adult and then left to swing. Assuming that each child on a swing can be treated as a simple pendulum and that friction is negligible, which child takes the longer time for one complete swing (has a longer period)? a) the bigger child d) the child given the b) the lighter child biggest push c) neither child

Short Answer

Expert verified
Answer: (c) Neither child has a longer period.

Step by step solution

01

Understand what a simple pendulum is

A simple pendulum consists of a small, heavy object (the “pendulum bob”) suspended by a light, inextensible string. The motion of the pendulum is determined by the length of the string and the angle it makes with the vertical, called the displacement angle.
02

Derive the equation for the period of a simple pendulum

The equation for the period T (the time it takes for one full oscillation) of a simple pendulum can be derived from the equation of motion using the small angle approximation. The equation for the period of a simple pendulum is given by: T = 2\pi\sqrt{\frac{L}{g}} where L is the length of the pendulum, and g is the acceleration due to gravity.
03

Identify the relevant parameters from the exercise

The exercise asks about the period of a pendulum that is affected by two factors: the mass of the child and the initial push given. According to the equation in step 2, the period T of a simple pendulum depends only on the length L and acceleration due to gravity g. The mass of the child and the initial push are not included in the equation.
04

Determine which child has a longer period

Since the period of a simple pendulum does not depend on its mass or the initial push given, both children will have the same period, assuming they are on swings of the same height (length). Thus, neither child has a longer period. #Conclusion# The correct answer is (c): neither child has a longer period, as the period of a simple pendulum depends only on its length and the acceleration due to gravity, and not on the mass of the child or the initial push given.

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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?

What is the period of a simple pendulum that is \(1.00 \mathrm{~m}\) long in each situation? a) in the physics lab b) in an clevator accelerating at \(2.10 \mathrm{~m} / \mathrm{s}^{2}\) upward c) in an elevator accelerating \(2.10 \mathrm{~m} / \mathrm{s}^{2}\) downward d) in an elevator that is in free fall

Pendulum A has a bob of mass \(m\) hung from a string of length \(I_{i}\) pendulum \(B\) is identical to \(A\) except its bob has mass \(2 m\). Compare the frequencies of small oscillations of the two pendulums.

If you kick a harmonic oscillator sharply, you impart to it an initial velocity but no initial displacement. For a weakly damped oscillator with mass \(m\), spring constant \(k\). and damping force \(F_{y}=-b v,\) find \(x(t),\) if the total impulse delivered by the kick is \(J_{0}\).

Mass-spring systems and pendulum systems can both be used in mechanical timing devices. What are the advantages of using one type of system rather than the othes in a device designed to generate reproducible time measurements over an extended period of time?
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