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Chapter 9: Problem 4

A student measures the maximum speed of a block undergoing simple harmonic oscillations of amplitude \(A\) on the end of an ideal spring. If the block is replaced by one with twice its mass but the amplitude of its oscillations remains the same, then the maximum speed of the block will (A) decrease by a factor of 4 (B) decrease by a factor of 2 (C) decrease by a factor of \(\sqrt{2}\) (D) increase by a factor of 2

Short Answer

Expert verified
The maximum speed of the block will decrease by a factor of \(\sqrt{2}\), so the correct answer is (C).

Step by step solution

01

Understand the Problem

In a simple harmonic motion, an ideal spring with a block attached undergoes oscillations where maximum speed is given when the spring is at equilibrium position, that is, \(x = 0\). This speed is given by the formula \(v = \sqrt{k/m} * A\), where \(v\) is maximum speed, \(k\) is spring's constant, \(m\) is mass of block and \(A\) is amplitude of oscillation.
02

Consider the Impact of Changing Mass

As in exercise, the mass \(m\) is changed to \(2m\) while keeping the amplitude \(A\) and spring constant \(k\) same. The new maximum speed \(v_2 = \sqrt{k / 2m} * A = v / \sqrt{2}\), where \(v\) is the original maximum speed.
03

Determine the Factor of Change

By comparing new maximum speed and original speed, it's evident that maximum speed of the block decreases by a factor of \(\sqrt{2}\).

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Most popular questions from this chapter

A linear spring of force constant \(k\) is used in a physics lab experiment. A block of mass \(m\) is attached to the spring and the resulting frequency, \(f\), of the simple harmonic oscillations is measured. Blocks of various masses are used in different trials, and in each case, the corresponding frequency is measured and recorded. If \(f^{2}\) is plotted versus \(1 / m,\) the graph will be a straight line with slope (A) \(\frac{4 \pi^{2}}{k^{2}}\) (B) \(\frac{4 \pi^{2}}{k}\) (C) \(4 \pi^{2} k\) (D) \(\frac{k}{4 \pi^{2}}\).

A block with a mass of \(20 \mathrm{~kg}\) is attached to a spring with a force constant \(k=50 \mathrm{~N} / \mathrm{m}\). What is the magnitude of the acceleration of the block when the spring is stretched \(4 \mathrm{~m}\) from its equilibrium position? (A) \(4 \mathrm{~m} / \mathrm{s}^{2}\) (B) \(6 \mathrm{~m} / \mathrm{s}^{2}\) (C) \(8 \mathrm{~m} / \mathrm{s}^{2}\) (D) \(10 \mathrm{~m} / \mathrm{s}^{2}\)

A block attached to an ideal spring undergoes simple harmonic motion. The acceleration of the block has its maximum magnitude at the point where (A) the speed is the maximum (B) the speed is the minimum (C) the restoring force is the minimum (D) the kinetic energy is the maximum

A block attached to an ideal spring undergoes simple harmonic motion about its equilibrium position \((\mathrm{x}=\mathrm{o})\) with amplitude A. What fraction of the total energy is in the form of kinetic energy when the block is at position \(x=\frac{1}{2} A ?\) (A) \(\frac{1}{3}\) (B) \(\frac{1}{2}\) (C) \(\frac{2}{3}\) (D) \(\frac{3}{4}\)

A simple pendulum swings about the vertical equilibrium position with a maximum angular displacement of \(5^{\circ}\) and period \(T\). If the same pendulum is given a maximum angular displacement of \(10^{\circ}\), then which of the following best gives the period of the oscillations? A) \(\frac{T}{2}\) B) \(\frac{T}{\sqrt{2}}\) C) \(T\) D) \(2 T\).
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