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Chapter 10: Problem 72

A CD has a mass of \(15.0 \mathrm{~g}\), an inner diameter of \(1.5 \mathrm{~cm},\) and an outer diameter of \(11.9 \mathrm{~cm} .\) Suppose you toss it, causing it to spin at a rate of 4.3 revolutions per second. a) Determine the moment of inertia of the \(\mathrm{CD}\), approximating its density as uniform. b) If your fingers were in contact with the CD for 0.25 revolutions while it was acquiring its angular velocity and applied a constant torque to it, what was the magnitude of that torque?

Short Answer

Expert verified
\(I = \frac{1}{2}(15)(0.00005625 + 0.00354025)\) \(I = 7.5(0.0035965)\) \(I = 0.027\mathrm{~kg\cdot m^2}\) The moment of inertia of the CD is approximately \(0.027\mathrm{~kg\cdot m^2}\).

Step by step solution

01

a) Determine the moment of inertia of the CD

First, we need the formula for the moment of inertia of a thin circular ring, which is given by: \(I = \frac{1}{2}m(R_{1}^{2} + R_{2}^{2})\), where \(I\) is the moment of inertia, \(R_{1}\) is the inner radius, \(R_{2}\) is the outer radius, and \(m\) is the mass. Given that the inner diameter is \(1.5\mathrm{~cm}\), we can find the inner radius: \(R_{1} = \frac{1.5}{2} = 0.75\mathrm{~cm} =0.0075\mathrm{~m}\). Similarly, the outer diameter is \(11.9\mathrm{~cm}\), so the outer radius is \(R_{2} = \frac{11.9}{2} = 5.95\mathrm{~cm} = 0.0595\mathrm{~m}\). Now, we can plug the values into the formula to calculate the moment of inertia: \(I = \frac{1}{2}(15)(0.0075^2 + 0.0595^2)\)

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Most popular questions from this chapter

A professor doing a lecture demonstration stands at the center of a frictionless turntable, holding 5.00 -kg masses in each hand with arms extended so that each mass is \(1.20 \mathrm{~m}\) from his centerline. A (carefully selected!) student spins the professor up to a rotational speed of \(1.00 \mathrm{rpm} .\) If he then pulls his arms in by his sides so that each mass is \(0.300 \mathrm{~m}\) from his centerline, what is his new rotation rate? Assume that his rotational inertia without the masses is \(2.80 \mathrm{~kg} \mathrm{~m} / \mathrm{s}\), and neglect the effect on the rotational inertia of the position of his arms, since their mass is small compared to the mass of the body.

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