Q: A spherical shell of mass $$M = 820\ \mathrm{kg}$$ and radius $$R = 0.220\ \mathrm{m}$$ starts rotating from rest with an angular acceleration $$\alpha = 0.870\ \mathrm{$$rad/s^2$$}. $$What is the kinetic energy after $$6$$ revolutions?

$$KE = \frac{2}{3}MR^2$\Delta \theta$$

Q: If a mass $$m$$ falls a height $$h$$ while attached to a pulley of mass $$M$$ and radius $$R$$, which conservation law is most useful for finding the final speed?

Conservation of energy: $$KE + U = \text{constant}$$

Q: A spherical shell has a moment of inertia $$I = \frac{2}{3}MR^2$. If it starts from rest and rotates with angular acceleration $$\alpha$$, what is its angular velocity after rotating through angle $$\Delta \theta$$?

$$\omega = \sqrt{2\alpha \Delta \theta}$$

Question 1

A solid cylinder of mass $$M = 50\ \mathrm{kg}$$ and diameter $$D = 0.120\ \mathrm{m} is $$pulled by a force $$F = 9.0\ \mathrm{N}$$ over a distance $$d = 2.0\ \mathrm{m}. $$Assuming no friction and that the system starts from rest, what is the final angular velocity $$\omega of $$the cylinder?

Accepted Answer

$$\omega = \frac{2\sqrt{Fd}}{R\sqrt{M}}$$

Question 2

Which equation correctly expresses the parallel axis theorem for the moment of inertia?

Accepted Answer

$$I = I_{\mathrm{CM}} + Md^2$

Question 3

A uniform rod of length $$L$$ and mass $$M$$ has its moment of inertia about an axis through its center of mass given by $$I_{\mathrm{CM}} = \frac{1}{12}ML^2$. What is its moment of inertia about an axis at one end perpendicular to the rod?

Accepted Answer

$$I = \frac{1}{3}ML^2$

Question 4

A mass $$m is $$released from rest at a height $$h$$ above the ground, attached to a pulley of mass $$M$$ and radius $$R. $$What is the speed $$v of$$ $$m$$ just before it hits the ground?

Accepted Answer

$$v = \sqrt{\frac{2gh}{2 + \frac{M}{m}}}$$

Question 5

For the special case where the mass of the pulley $$M is $$much less than the mass $$m$$ ($$M \ll m$$), what does the speed $$v of $$the falling mass approach?

Accepted Answer

$$v \approx \sqrt{2gh}$$

Question 6

A spherical shell of mass $$M = 820\ \mathrm{kg}$$ and radius $$R = 0.220\ \mathrm{m}$$ starts rotating from rest with an angular acceleration $$\alpha = 0.870\ \mathrm{$$rad/s^2$$}. $$What is the kinetic energy after $$6$$ revolutions?

Accepted Answer

$$KE = \frac{2}{3}MR^2$\Delta 	heta$$

Question 7

If a force $$F$$ acts tangentially on a solid cylinder of radius $$R$$, what is the torque $$	au$$ produced?

Accepted Answer

$$	au = FR$$

Question 8

Which of the following best describes the work-energy theorem as applied to rotational motion?

Accepted Answer

$$W = \Delta KE = \frac{1}{2}I\$$omega^2$$ - \frac{1}{2}I\$$omega_0$$^2$$

Question 9

A solid cylinder of mass $$M$$ and radius $$R$$ rotates with angular velocity $$\omega. $$What is its rotational kinetic energy?

Accepted Answer

$$KE = \frac{1}{2}I\$$omega^2$$ = \frac{1}{4}MR^2$\$$omega^2$$

Question 10

A uniform rod of mass $$M$$ and length $$L is $$pivoted at one end and released from rest in a horizontal position. What is its moment of inertia about the pivot?

Accepted Answer

$$I = \frac{1}{3}ML^2$

Question 11

If a mass $$m$$ falls a height $$h$$ while attached to a pulley of mass $$M$$ and radius $$R$$, which conservation law is most useful for finding the final speed?

Accepted Answer

Conservation of energy: $$KE + U = 	ext{constant}$$

Question 12

A spherical shell has a moment of inertia $$I = \frac{2}{3}MR^2$. If it starts from rest and rotates with angular acceleration $$\alpha$$, what is its angular velocity after rotating through angle $$\Delta 	heta$$?

Accepted Answer

$$\omega = \sqrt{2\alpha \Delta 	heta}$$

Question 13

Which of the following is the correct expression for the moment of inertia of a solid cylinder about its central axis?

Accepted Answer

$$I = \frac{1}{2}MR^2$

Question 14

A force $$F is $$applied tangentially to a solid cylinder of mass $$M$$ and radius $$R. $$What is the resulting angular acceleration $$\alpha$$?

Accepted Answer

$$\alpha = \frac{2F}{MR}$$

Question 15

If a rod of mass $$M$$ and length $$L is $$rotated about an axis a distance $$d$$ from its center of mass, what is its moment of inertia?

Accepted Answer

$$I = I_{\mathrm{CM}} + Md^2$

Question 16

A mass $$m$$ falls attached to a pulley of mass $$M$$ and radius $$R. If$$ $$M \gg m$$, what is the approximate final speed $$v of $$the mass?

Accepted Answer

$$v \approx \sqrt{\frac{gh}{2}}$$

Question 17

Which of the following is a correct statement about energy conservation in a system with both translational and rotational motion?

Accepted Answer

$$mgh = \frac{1}{2}mv^2$ + \frac{1}{2}I\$$omega^2$$

Rotational Dynamics, Energy, and the Parallel Axis Theorem

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