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Chapter 13: Problem 30

An object executes simple harmonic motion with an amplitude \(A\). (a) At what values of its position does its speed equal half its maximum speed? (b) At what values of its position does its potential energy equal half the total energy?

Short Answer

Expert verified
The speed of the object is half its maximum speed at the positions \( 卤 \frac{A}{\sqrt{2}} \) and the potential energy equals half the total energy at the positions \(卤A\).

Step by step solution

01

Compute the position for half maximum speed

First, we equate the speed \(v = \omega \sqrt{A^{2}-x^{2}}\) to half the maximum speed, i.e., \(\frac{\omega \cdot A}{2}\) and solve for \(x\) to get the positions where the speed equals half its maximum speed.
02

Simplify the result

After cancelling \(\omega\) and squaring both sides, we get \( A^{2} - x^{2} = \frac{A^{2}}{4}\) . We solve this equation for \(x\) and simplify the result to obtain \(x = 卤 \frac{A}{\sqrt{2}}\)
03

Compute the position for half total energy

Now, for part (b), we equate the potential energy \(U = \frac{1}{2} m \omega^{2} x^{2}\) to half the total energy, i.e., \(\frac{1}{4} m \omega^{2} A^{2} \) and solve for \(x\)
04

Simplify the final result

After cancelling \(m\), \(\omega^{2}\), and simplifying the equation, we get \(x = 卤 A\). This means that the potential energy equals half the total energy at both extremes, i.e., the maximum and minimum position.

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Most popular questions from this chapter

A vertical spring stretches \(3.9 \mathrm{~cm}\) when a \(10-\mathrm{g}\) object is hung from it. The object is replaced with a block of mass \(25 \mathrm{~g}\) that oscillates up and down in simple harmonic motion. Calculate the period of motion.

A bat can detect small objects, such as an insect, whose size is approximately equal to one wavelength of the sound the bat makes. If bats emit a chirp at a frequency of \(60.0 \times 10^{3} \mathrm{~Hz}\) and the speed of sound in air is \(343 \mathrm{~m} / \mathrm{s}\), what is the smallest insect a bat can detect?

A wave of amplitude \(0.30 \mathrm{~m}\) interferes with a second wave of amplitude \(0.20 \mathrm{~m}\) traveling in the same direction. What are (a) the largest and (b) the smallest resultant amplitudes that can occur, and under what conditions will these maxima and minima arise?

A \(2.00-\mathrm{kg}\) block hangs without vibrating at the end of a spring \((k=500 \mathrm{~N} / \mathrm{m})\) that is attached to the ceiling of an elevator car. The car is rising with an upward acceleration of \(g / 3\) when the acceleration suddenly ceases (at \(t=0\) ). (a) What is the angular frequency of oscillation of the block after the acceleration ceases? (b) By what amount is the spring stretched during the time that the elevator car is accelerating?

A load of \(50 \mathrm{~N}\) attached to a spring hanging vertically stretches the spring \(5.0 \mathrm{~cm}\). The spring is now placed horizontally on a table and stretched \(11 \mathrm{~cm}\). (a) What force is required to stretch the spring by that amount? (b) Plot a graph of force (on the \(y\)-axis) versus spring displacement from the equilibrium position along the \(x\)-axis.
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