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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
After differentiation and substituting the velocity definition, we get \(0 = m \frac{d^{2} x}{d t^{2}} + k x\). This is a standard expression for a harmonic oscillator, which is assumed to be Equation 13.3.

Step by step solution

01

Write down the expression for total energy

Given the total energy \(E\) of the system is equal to the sum of kinetic and potential energies and it remains constant, this gives the equation \(E=\frac{1}{2} m v^{2}+\frac{1}{2} k x^{2}\).
02

Differentiate both sides of the energy equation

Differentiating both sides with respect to time, we get \(\frac{d E}{d t} = m v \frac{d v}{d t} + k x \frac{d x}{d t}\). Since \(E\) is a constant, \(\frac{d E}{d t} = 0\). Therefore, we get \(0 = m v \frac{d v}{d t} + k x \frac{d x}{d t}\).
03

Use the definition of velocity

As noted in the hint, the velocity \(v\) is defined as \(v = \frac{d x}{d t}\). Substituting this into the obtained equation yields \(0 = m (\frac{d x}{d t}) \frac{d (\frac{d x}{d t})}{d t} + k x \frac{d x}{d t}\), simplifying gives \(0 = m \frac{d^{2} x}{d t^{2}} + k x\).

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Most popular questions from this chapter

A pendulum of length \(L\) is mounted in a rocket. Find its period if the rocket is (a) at rest on its launch pad; (b) accelerating upward with acceleration \(a=\frac{1}{2} g ;(c)\) accelerating downward with \(a=\frac{1}{2} g ;\) and \((d)\) in free fall.

A \(500-\mathrm{g}\) mass is suspended from a thread \(45 \mathrm{cm}\) long that can sustain a tension of \(6.0 \mathrm{N}\) before breaking. Find the maximum allowable amplitude for pendulum motion of this system.

A simple model for a variable star considers that the outer layer of the star is subject to two forces: the inward force of gravity and the outward force due to gas pressure. As a result, Newton's law for the star's outer layer reads \(m d^{2} r / d t^{2}=4 \pi r^{2} p-G M m / r^{2} .\) Here \(m\) is the mass of the outer layer, \(M\) is the total mass of the star, \(r\) is the star's radius, and \(p\) is the pressure. (a) Use this equation to show that the star's equilibrium pressure and radius are related by \(p_{0}=G M m / 4 \pi r_{0}^{4},\) where the subscript 0 represents equilibrium values. (b) As you'll learn in Chapter 18 , gas pressure and volume \(V\left(=\frac{4}{3} \pi r^{3}\right)\) are related by \(p V^{3 / 3}=p_{0} V_{0}^{5 / 3}\) (this is for an adiabatic process, a good approximation here, and the exponent \(5 / 3\) reflects the ionized gas that makes up the star). Let \(x=r-r_{0}\) be the displacement of the star's surface from equilibrium. Use the binomial approximation (Appendix A) to show that, when \(x\) is small compared with \(r,\) the righthand side of the above equation can be written \(-\left(G M m / r_{0}^{3}\right) x\) (c) since \(r\) and \(x\) differ only by a constant, the term \(d^{2}\) r/dt \(^{2}\) in the equation above can also be written \(d^{2} x / d t^{2} .\) Make this substitution, along with substituting the result of part (b) for the right- hand side, and compare your result with Equations 13.2 and 13.7 to find an expression for the oscillation period of the star. (d) What does your simple model predict for the period of the variable star Delta Cephei, with radius 44.5 times that of the Sun and mass of 4.5 Sun masses? (Your answer overestimates the actual period by a factor of about \(3,\) both because of oversimplified physics and because changes in the star's radius are too large for the assumption of a linear restoring force.)

A 200 -g mass is attached to a spring of constant \(k=5.6 \mathrm{N} / \mathrm{m}\) and set into oscillation with amplitude \(A=25 \mathrm{cm} .\) Determine (a) the frequency in hertz, (b) the period, (c) the maximum velocity, and (d) the maximum force in the spring.

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