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Chapter 13: Problem 62

The total energy of a mass-spring system is the sum of its kinetic and potential energy: \(E=\frac{1}{2} m v^{2}+\frac{1}{2} k x^{2} .\) Assuming \(E\) remains constant, differentiate both sides of this expression with respect to time and show that Equation 13.3 results. (Hint: Remember that \(v=d x / d t .)\)

Short Answer

Expert verified
By differentiating the given energy equation with respect to time and substituting \(v = \frac{dx}{dt}\) into the resulting expression, we derive Equation 13.3, which is \(\frac{dE}{dt} = m v^{2} + kx v\).

Step by step solution

01

Write Down the given Equation

The total energy of a mass-spring system is given by \(E=\frac{1}{2} m v^{2}+\frac{1}{2} k x^{2}\). This is the equation that will be differentiated.
02

Differentiate both sides with respect to time

Differentiating both sides of the equation with respect to time leads to \(\frac{dE}{dt} = m v \frac{dv}{dt} + kx \frac{dx}{dt}\)
03

Apply the hint

The hint tells us that \(v = \frac{dx}{dt}\). Substituting into our differentiation result from step 2, we have \(\frac{dE}{dt} = m v^{2} + kx \frac{dx}{dt}\). But as noted, \(v = \frac{dx}{dt}\), so the equation simplifies to \(\frac{dE}{dt} = m v^{2} + kx v\), which is the form of Equation 13.3. Hence, by differentiating the given equation, we have derived Equation 13.3.

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Most popular questions from this chapter

Derive the period of a simple pendulum by considering the horizontal displacement \(x\) and the force acting on the bob, rather than the angular displacement and torque.

A particle of mass \(m\) has potential energy given by \(U=a x^{2}\) where \(a\) is a constant and \(x\) is the particle's position. Find an expression for the frequency of simple harmonic oscillations this particle undergoes.

A 50 -g mass is attached to a spring and undergoes simple harmonic motion. Its maximum acceleration is \(15 \mathrm{m} / \mathrm{s}^{2}\) and its maximum speed is \(3.5 \mathrm{m} / \mathrm{s}\). Determine the (a) angular frequency, (b) spring constant, and (c) amplitude.

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