Q: Which equation correctly relates angular displacement $$\Delta \theta$$, initial angular velocity $$\$$omega_0$$, angular acceleration $$\alpha$$, and time t$ for constant $$\alpha$$?

$$\Delta \theta = \$$omega_0$$ t + \frac{1}{2} \alpha t^2$

Q: A wheel starts from rest and accelerates at $$2\ \mathrm{$$rad/s^2$$}$$ for 4 seconds. What is its angular displacement?

$$\Delta \theta = 16\ \mathrm{rad}$$

Question 1

A barbell consists of a 0.5 kg ball at (0,0) and a 2 kg ball at (0.5,0) meters, connected by a massless rod. What is the x-coordinate of the center of mass ($$x_{CM}) of $$the system?

Accepted Answer

$$x_{CM} = 0.4\ \mathrm{m}$$

Question 2

If the masses in the previous barbell problem are swapped (2 kg at (0,0) and 0.5 kg at (0.5,0)), what is the new x-coordinate of the center of mass ($$x_{CM}$$)?

Accepted Answer

$$x_{CM} = 0.1\ \mathrm{m}$$

Question 3

A barbell has a 2 kg ball (radius 10 cm) and a 0.5 kg ball (radius 5 cm) connected by a 50 cm rod attached at their edges. What is the x-coordinate of the center of mass ($$x_{CM}$$)?

Accepted Answer

$$x_{CM} = 0.52\ \mathrm{m}$$

Question 4

Now, if the rod in the previous problem has a mass of 0.1 kg distributed uniformly and is 0.5 m long, what is the x-coordinate of the center of mass ($$x_{CM}) of $$the entire system?

Accepted Answer

$$x_{CM} = 0.51\ \mathrm{m}$$

Question 5

A compact disk rotates at 500 rev/min. What is its angular velocity in radians per second?

Accepted Answer

$$\omega = 52.3\ \mathrm{rad/s}$$

Question 6

A compact disk with a diameter of 120 mm rotates at 500 rev/min. What is the tangential speed at the edge of the disk?

Accepted Answer

$$v = 3.1\ \mathrm{m/s}$$

Question 7

A wheel of radius 0.1 m has a constant angular acceleration of $$1.0\ \mathrm{$$rad/s^2$$}$$ and an initial angular velocity of $$2\ \mathrm{rad/s}. $$What is its angular velocity after 5 seconds?

Accepted Answer

$$\omega = 7\ \mathrm{rad/s}$$

Question 8

For the same wheel in the previous question, what is the tangential velocity of a point at the rim after 5 seconds?

Accepted Answer

$$v_t = 0.7\ \mathrm{m/s}$$

Question 9

Which of the following best describes the center of mass of a system?

Accepted Answer

It is the point where the total mass of the system can be considered to be concentrated for translational motion.

Question 10

A human body is modeled as two segments: upper (45 kg at (15,50) cm) and lower (30 kg at (30,20) cm). What is the y-coordinate of the center of mass?

Accepted Answer

$$y_{CM} = 38\ \mathrm{cm}$$

Question 11

Which equation correctly relates angular displacement $$\Delta 	heta$$, initial angular velocity $$\$$omega_0$$, angular acceleration $$\alpha$$, and time t$ for constant $$\alpha$$?

Accepted Answer

$$\Delta 	heta = \$$omega_0$$ t + \frac{1}{2} \alpha t^2$

Question 12

If a point on a rotating disk is halfway to the edge (radius $$r = 0.03\ \mathrm{m}) $$and the disk spins at $$\omega = 52.3\ \mathrm{rad/s}$$, what is the tangential speed at that point?

Accepted Answer

$$v = 1.6\ \mathrm{m/s}$$

Question 13

Which of the following statements about rotational motion is correct?

Accepted Answer

All points on a rigid body rotating about a fixed axis have the same angular velocity.

Question 14

A wheel starts from rest and accelerates at $$2\ \mathrm{$$rad/s^2$$}$$ for 4 seconds. What is its angular displacement?

Accepted Answer

$$\Delta 	heta = 16\ \mathrm{rad}$$

Question 15

Which formula gives the relationship between tangential speed $$v$$ and angular velocity $$\omega$$ for a point at radius $$r$$?

Accepted Answer

$$v = \omega r$$

Question 16

A gymnast is balancing horizontally on a bar. If her upper body (45 kg) is centered at (15,50) cm and her lower body (30 kg) at (30,20) cm, what is the x-coordinate of her center of mass?

Accepted Answer

$$x_{CM} = 21\ \mathrm{cm}$$

Question 17

Which of the following is NOT a correct statement about the center of gravity?

Accepted Answer

It is always located at the geometric center of the object.

Rotation I: Center of Mass and Rotational Kinematics

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