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Chapter 13: Problem 34

A \(450-\mathrm{g}\) mass on a spring is oscillating at \(1.2 \mathrm{Hz},\) with total energy \(0.51 \mathrm{J} .\) What's the oscillation amplitude?

Short Answer

Expert verified
The amplitude of the oscillation is 0.19 m.

Step by step solution

01

Convert given parameters into correct units and identify the values.

First, convert the given mass from grams to kilograms, as the SI unit of mass is kilogram. Therefore, \(m = 450 g = 0.45 kg.\) The given frequency of oscillation is \(f = 1.2 Hz\), and the amplitude of oscillation \(A\) is what we're aiming to find. The total energy in the system is given as \(E = 0.51 J\). Remember to always keep quantities in their respective SI units during calculations.
02

Compute the angular frequency \(w\).

The angular frequency \(w\) can be calculated from the given frequency using the formula \(w = 2 \pi f\). Substituting the given values into the equation, \(w = 2 \pi (1.2 Hz) = 7.54 rad/s\).
03

Solve for amplitude.

We already know that the total energy in an oscillating system can be written as \(E = \frac{1}{2} m w^2 A^2\). To find the amplitude \(A\), we can rearrange this equation to the form \(A = \sqrt{\frac{2E}{m w^2}}\). Substituting the given and calculated values, we get \(A = \sqrt{\frac{2(0.51 J)}{0.45 kg (7.54 rad/s)^2}} = 0.19 m\).
04

Interpret the Results

The calculated amplitude indicates the maximum displacement from the equilibrium position. Therefore, the mass on the spring oscillates 0.19 m away from its equilibrium position at maximum displacement.

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Most popular questions from this chapter

How can a system have more than one resonant frequency?

Two identical mass-spring systems consist of \(430-\mathrm{g}\) masses on springs of constant \(k=2.2 \mathrm{N} / \mathrm{m} .\) Both are displaced from equilibrium, and the first is released at time \(t=0 .\) How much later should the second be released so their oscillations differ in phase by \(\pi / 2 ?\)

Is a vertically bouncing ball an example of oscillatory motion? Of simple harmonic motion? Explain.

A particle undergoes simple harmonic motion with maximum speed \(1.4 \mathrm{m} / \mathrm{s}\) and maximum acceleration \(3.1 \mathrm{m} / \mathrm{s}^{2} .\) Find the (a) angular frequency, (b) period, and (c) amplitude.

How does the frequency of a simple harmonic oscillator depend on its amplitude?
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