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Chapter 4: Problem 17

A figure skater can be idealized as composed of two solid cylinders of uniform density crossed at right angles to one another. Assume that the skater is spinning with angular speed \(\omega\) around a vertical axis passing through the center of her body. The mass of the large cylinder is \(M\); it has radius \(R\) and length \(L\). The mass of the small cylinder is \(m\); its radius is \(r\) and its length is \(\ell\). a) What is the moment of inertia of the large vertical cylinder in terms of the given quantities? b) What is the moment of inertia of the horizontal cylinder in terms of the given quantities? c) What is the figure skater's initial angular momentum? d) If \(m=0.1 M, \ell=8 R\) and the skater's arms can be totally retracted to her body, what is her new angular speed in terms of \(\omega\) ?

Short Answer

Expert verified
In order to calculate the moment of inertia for the large vertical cylinder, we must first identify the mass (M) and radius (R) of the cylinder. Let's assume the mass of the large vertical cylinder is M1 and its radius is R1. Then, the moment of inertia for the large vertical cylinder (I1) can be calculated using the formula: I1 = (1/2)M1R1^2

Step by step solution

01

Calculate moment of inertia for the large vertical cylinder

For a solid cylinder rotating about a central axis, its moment of inertia I is given by the formula: \(I = \frac{1}{2}MR^2\). This is the moment of inertia of the large vertical cylinder.

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Most popular questions from this chapter

A gyroscope is essentially a spinning wheel attached to an axle that is free to rotate in any direction about a pivot. In the figure below, a gyro wheel of mass \(M\) and radius \(R\) spins with angular velocity \(\omega\) in a counterclockwise direction when viewed from the right. The distance between the wheel and the pivot \(O\) is \(D\). Assume that the axle is of negligible mass and that the wheel's entire mass is concentrated around the rim. a) Is there an axis of symmetry in the problem? If so what is it? b) What is the magnitude of the angular momentum of the gyroscope? Draw the direction the angular momentum vector is pointing on the diagram. c) Are there any external forces acting on the gyro wheel? If so, draw them on the diagram. d) What is the magnitude of any torque acting on the wheel? Draw the direction of the torque on the diagram. e) Does the gyroscope move in any direction? If so which?

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A bullet of mass \(m\) is fired with a velocity \(v\) into a wooden disk of mass \(M\) and radius \(R\) that is free to rotate, as shown. The bullet lodges itself in the disk at a height \(b\) above the wheel's center and the system begins to rotate with an angular velocity \(\omega\). Assume \(M \gg m\). In order to predict \(\omega\), one needs (A) conservation of energy. (B) conservation of momentum. (C) conservation of angular momentum. (D) conservation of energy and conservation of momentum. (E) conservation of energy and conservation of angular momentum.

A window washer of \(M=75 \mathrm{~kg}\) is taking a lunch break and sitting \(4 / 3 \mathrm{~m}\) from the left edge of a 5- meter-long board that hangs from two cables. The mass \(m\) of the board is \(20 \mathrm{~kg}\). If \(T_1\) is the tension in the right cable and \(T_2\) is the tension in the left cable, what are their values? (A) \(T_1=200 \mathrm{~N} ; T_2=650 \mathrm{~N}\) (B) \(T_1=300 \mathrm{~N} ; T_2=650 \mathrm{~N}\) (C) \(T_1=650 \mathrm{~N} ; T_2=300 \mathrm{~N}\) (D) \(T_1=650 \mathrm{~N} ; T_2=200 \mathrm{~N}\) (E) \(T_1=450 \mathrm{~N} ; T_2=400 \mathrm{~N}\)
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