Q: A centrifuge spins at $$12,000\ \mathrm{rpm}$$ with a sample $$5\ \mathrm{cm}$$ from the axis. What is the centripetal acceleration $$a_{cp} at $$the sample location? (Use $$1\ \mathrm{rpm} = \dfrac{2\pi}{60}\ \mathrm{rad/s})$$

$$a_{cp} = r\$$omega^2$$ = 0.05\ \mathrm{m} \times (1256.6\ \mathrm{rad/s})^2$ \approx 78,952\ \mathrm{m/s^2$}$$

Q: A platform accelerates from rest at $$\alpha = 0.01\ \mathrm{$$rad/s^2$$}. $$Ninja B stands at $$r = 20\ \mathrm{m}. $$How far (in linear distance) does Ninja B travel after $$10\ \mathrm{s}$$?

$$s = r \left(\dfrac{1}{2}\alpha t^2$\right) = 20 \times 0.5 = 10\ \mathrm{m}$$

Question 1

A point P is located at a distance $$r$$ from the center of a spinning disk. If the disk rotates through an angle $$	heta (in $$radians), what is the linear distance $$s$$ traveled by point P?

Accepted Answer

$$s = r	heta$$

Question 2

Two ninja are standing at different distances from the center of a rotating turntable. Ninja A is at $$8.0\ \mathrm{m}$$ and Ninja B is at $$20.0\ \mathrm{m}. $$The turntable rotates once every $$12.57\ \mathrm{s}. $$What is the angular speed $$\omega of $$the turntable?

Accepted Answer

$$\omega = \dfrac{2\pi}{12.57\ \mathrm{s}} \approx 0.5\ \mathrm{rad/s}$$

Question 3

A car is traveling at $$30\ \mathrm{m/s}$$ and its tires have a radius of $$0.3\ \mathrm{m}. $$What is the angular speed $$\omega of $$the tires?

Accepted Answer

$$\omega = \dfrac{v}{r} = \dfrac{30}{0.3} = 100\ \mathrm{rad/s}$$

Question 4

A centrifuge spins at $$12,000\ \mathrm{rpm}$$ with a sample $$5\ \mathrm{cm}$$ from the axis. What is the centripetal acceleration $$a_{cp} at $$the sample location? (Use $$1\ \mathrm{rpm} = \dfrac{2\pi}{60}\ \mathrm{rad/s})$$

Accepted Answer

$$a_{cp} = r\$$omega^2$$ = 0.05\ \mathrm{m} 	imes (1256.6\ \mathrm{rad/s})^2$ \approx 78,952\ \mathrm{m/s^2$}$$

Question 5

A solid disk of mass $$M$$ and radius $$R$$ rotates about its center. A small mass $$m is $$placed on the rim. What is the total moment of inertia of the system?

Accepted Answer

$$I = \dfrac{1}{2}MR^2$ + mR^2$

Question 6

A hoop and a disk, both with mass $$M$$ and radius $$R$$, roll down an incline from rest. Which reaches the bottom first and why?

Accepted Answer

The disk, because it has a smaller moment of inertia and more energy goes into translational motion.

Question 7

A wheel of radius $$r$$ rolls without slipping. What is the velocity of the point at the top of the wheel relative to the ground?

Accepted Answer

$$v_{top} = 2r\omega$$

Question 8

A small wheel and a large wheel are connected by a belt. The small wheel rotates at angular velocity $$\omega_S. If $$the belt does not slip, how does the angular velocity $$\omega_L of $$the large wheel compare?

Accepted Answer

$$\omega_L \u003c \omega_S$$

Question 9

A platform accelerates from rest at $$\alpha = 0.01\ \mathrm{$$rad/s^2$$}. $$Ninja B stands at $$r = 20\ \mathrm{m}. $$How far (in linear distance) does Ninja B travel after $$10\ \mathrm{s}$$?

Accepted Answer

$$s = r \left(\dfrac{1}{2}\alpha t^2$ight) = 20 	imes 0.5 = 10\ \mathrm{m}$$

Question 10

A solid sphere, a solid disk, and a hoop all have the same mass $$M$$ and radius $$R$$ and roll without slipping at the same speed $$v. $$Which has the largest total kinetic energy?

Accepted Answer

The hoop

Question 11

A rolling object has both translational and rotational kinetic energy. For a rolling object with $$I = \beta MR^2$, what is the total kinetic energy in terms of v$?

Accepted Answer

$$K = \dfrac{1}{2}Mv^2$(1 + \beta)$$

Question 12

A ladybug sits on a turntable that is spinning and slowing down (viewed from above). Which of the following best describes the angular acceleration $$\alpha$$ and angular velocity $$\omega$$?

Accepted Answer

$$\alpha  0$$

Question 13

What is the total magnitude of acceleration for Ninja B after $$t = 10\ \mathrm{s}$$, given $$\alpha = 0.01\ \mathrm{$$rad/s^2$$}$$ and $$r = 20\ \mathrm{m}$$?

Accepted Answer

$$a_{TOT} = \sqrt{a_{cp}^2 + a_T^2$} = \sqrt{0.2^2$ + 0.2^2$} = 0.28\ \mathrm{m/s^2$}$$

Question 14

A baseball bat is modeled as a uniform rod of length $$L$$ and mass $$M. $$What is its moment of inertia about an axis through one end perpendicular to its length?

Accepted Answer

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

Question 15

A solid sphere of mass $$M$$ and radius $$R$$ rolls without slipping. What fraction of its total kinetic energy is rotational?

Accepted Answer

$$\dfrac{2}{7}$$

Question 16

If the axis of rotation is shifted parallel to the center of mass by a distance $$d$$, what is the new moment of inertia $$I_p$$?

Accepted Answer

$$I_p = I_{cm} + Md^2$

Question 17

A block rolls along rollers without slipping, and the rollers move along the ground without slipping. If the block moves at speed $$v$$ and the rollers have radius $$R$$, what is the angular speed $$\omega of $$the rollers?

Accepted Answer

$$\omega = \dfrac{v}{2R}$$

Question 18

A NASA centrifuge has a radius of $$8.8\ \mathrm{m}$$ and is spun at $$35\ \mathrm{rpm}. $$What is the centripetal acceleration in terms of $$g$$ ($$g = 9.8\ \mathrm{m/s^2$}$$)?

Accepted Answer

$$a_{cp} = 12g$$

Rotational Kinematics and Energy: Mini-Textbook Study Notes

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