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Chapter 10: Problem 47

The displacement of an object in SHM is given by $y(t)=(8.0 \mathrm{cm}) \sin [(1.57 \mathrm{rad} / \mathrm{s}) t] .$ What is the frequency of the oscillations?

Short Answer

Expert verified
Answer: The frequency of the oscillations is approximately \(0.25\,\mathrm{Hz}\).

Step by step solution

01

Identify the equation of motion

The displacement of the object in SHM is given by \(y(t) = (8.0 \mathrm{cm})\sin[(1.57 \mathrm{rad} / \mathrm{s})t]\). This equation has the general form \(y(t) = A\sin(\omega t)\), where A is the amplitude and ω is the angular frequency.
02

Extract the angular frequency

From the given equation, we can see that the angular frequency ω is equal to \(1.57 \mathrm{rad} / \mathrm{s}\).
03

Find the frequency

We know the relation between angular frequency (ω) and frequency (f) is given as \(\omega = 2 \pi f\). We need to solve this equation for f. $$ f = \frac{\omega}{2 \pi} $$ Now, substitute ω with the value we found earlier: $$ f = \frac{1.57 \mathrm{rad}/\mathrm{s}}{2 \pi} $$ Calculate the frequency f: $$ f \approx 0.25\,\mathrm{Hz} $$ The frequency of the oscillations is approximately \(0.25\,\mathrm{Hz}\).

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Most popular questions from this chapter

A horizontal spring with spring constant of \(9.82 \mathrm{N} / \mathrm{m}\) is attached to a block with a mass of \(1.24 \mathrm{kg}\) that sits on a frictionless surface. When the block is 0.345 m from its equilibrium position, it has a speed of \(0.543 \mathrm{m} / \mathrm{s}\) (a) What is the maximum displacement of the block from the equilibrium position? (b) What is the maximum speed of the block? (c) When the block is \(0.200 \mathrm{m}\) from the equilibrium position, what is its speed?
What is the maximum load that could be suspended from a copper wire of length \(1.0 \mathrm{m}\) and radius \(1.0 \mathrm{mm}\) without permanently deforming the wire? Copper has an elastic limit of \(2.0 \times 10^{8} \mathrm{Pa}\) and a tensile strength of \(4.0 \times 10^{8} \mathrm{Pa}\).
A body is suspended vertically from an ideal spring of spring constant $2.5 \mathrm{N} / \mathrm{m}$. The spring is initially in its relaxed position. The body is then released and oscillates about its equilibrium position. The motion is described by $$y=(4.0 \mathrm{cm}) \sin [(0.70 \mathrm{rad} / \mathrm{s}) t]$$ What is the maximum kinetic energy of the body?
(a) What is the energy of a pendulum \((L=1.0 \mathrm{m}, m=\) $0.50 \mathrm{kg})\( oscillating with an amplitude of \)5.0 \mathrm{cm} ?$ (b) The pendulum's energy loss (due to damping) is replaced in a clock by allowing a \(2.0-\mathrm{kg}\) mass to drop \(1.0 \mathrm{m}\) in 1 week. What average percentage of the pendulum's energy is lost during one cycle?
A mass-and-spring system oscillates with amplitude \(A\) and angular frequency \(\omega\) (a) What is the average speed during one complete cycle of oscillation? (b) What is the maximum speed? (c) Find the ratio of the average speed to the maximum speed. (d) Sketch a graph of \(v_{x}(t),\) and refer to it to explain why this ratio is greater than \(\frac{1}{2}\).
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