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Chapter 10: Problem 39

Equipment to be used in airplanes or spacecraft is often subjected to a shake test to be sure it can withstand the vibrations that may be encountered during flight. A radio receiver of mass \(5.24 \mathrm{kg}\) is set on a platform that vibrates in SHM at \(120 \mathrm{Hz}\) and with a maximum acceleration of $98 \mathrm{m} / \mathrm{s}^{2}(=10 \mathrm{g}) .$ Find the radio's (a) maximum displacement, (b) maximum speed, and (c) the maximum net force exerted on it.

Short Answer

Expert verified
Solution: a) Maximum Displacement (x_max): x_max = 98 m/s² / (2π(120 Hz))² b) Maximum Speed (v_max): v_max = 2π(120 Hz) * x_max c) Maximum net force (F_net): F_net = 5.24 kg * 98 m/s²

Step by step solution

01

Find the angular frequency

First, we need to find the angular frequency (ω). We know the frequency of the vibration is 120 Hz, so we can convert this to angular frequency using the formula: ω = 2πf where f is the frequency. ω = 2π(120 Hz)
02

Calculate the maximum displacement

We are given the maximum acceleration amax = 98 m/s². Using the formula for acceleration in simple harmonic motion: a(t) = -ω²x(t) amax = ω²xmax, where xmax is the maximum displacement. So, xmax = amax / ω² xmax = 98 m/s² / (2π(120Hz))²
03

Calculate the maximum speed

Using the formula for velocity in simple harmonic motion: v(t) = ωx(t) vmax = ωxmax Using the value of xmax from step 2, we can calculate vmax: vmax = 2π(120 Hz) * xmax
04

Calculate the maximum net force

Using Newton's second law of motion, the maximum net force, Fnet, on the radio receiver is given by the product of its mass and the maximum acceleration: Fnet = m * amax Using the given mass m = 5.24 kg and amax = 98 m/s², we can find the maximum net force: Fnet = 5.24 kg * 98 m/s² Now, we can plug in the values to get the final results:
05

Final results

a) Maximum Displacement: x_max = 98 m/s² / (2π(120 Hz))² b) Maximum Speed: v_max = 2π(120 Hz) * x_max c) Maximum net force: F_net = 5.24 kg * 98 m/s²

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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?
A pendulum of length \(75 \mathrm{cm}\) and mass \(2.5 \mathrm{kg}\) swings with a mechanical energy of 0.015 J. What is the amplitude?
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A ball is dropped from a height \(h\) onto the floor and keeps bouncing. No energy is dissipated, so the ball regains the original height \(h\) after each bounce. Sketch the graph for \(y(t)\) and list several features of the graph that indicate that this motion is not SHM.
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