Q: A 3.2 kg flagpole is attached to a wall at an angle of $$25^\circ$$ from the horizontal. Its center of gravity is 1.6 m from the wall. What is the magnitude of the gravitational torque about the attachment point? ($$g = 9.8 m/s$^2)$$

$$\tau = (3.2 \times 9.8 \times 1.6) \sin 25^\circ$$

Q: Emma (25 kg) and Noah (22 kg) are on a seesaw. Noah sits 1.6 m from the pivot. How far from the pivot must Emma sit to balance the seesaw? (Assume seesaw mass is negligible.)

$$x_E = -\frac{22}{25} \times 1.6 m$$

Q: A flywheel (solid cylinder, $$M = 540 kg$$, $$R = 0.30 m) is $$spun up by a constant torque. If the final angular speed is $$\omega_f = 1360$$ rad/s, what is the moment of inertia $$I of $$the flywheel?

$$I = \frac{1}{2} \times 540 \times (0.30)^2$

Question 1

A wind turbine blade rotates about its central axis. If the tip of the blade is 40 m from the axis and the blade completes one revolution every 4 seconds, what is the linear speed $$v of $$the tip of the blade? Use $$v = r \omega.$$

Accepted Answer

$$v = 10\pi m/s$$

Question 2

A particle moves in a circle of radius $$r$$ such that it sweeps out an arc length $$s in $$time $$t. $$The angular position $$	heta (in $$radians) is given by which equation?

Accepted Answer

$$	heta = \frac{s}{r}$$

Question 3

A bicycle wheel accelerates from rest to an angular velocity of $$12$$ rad/s in $$3$$ seconds. What is its angular acceleration $$\alpha$$?

Accepted Answer

$$\alpha = 4$$ rad/s$$^2$$

Question 4

A door is pushed with a force $$F at a $$distance $$r$$ from the hinge, at an angle $$\phi to $$the radial line. What is the torque $$	au$$ about the hinge?

Accepted Answer

$$	au = rF\sin\phi$$

Question 5

A uniform rod of length $$L$$ and mass $$M$$ rotates about one end. What is its moment of inertia $$I$$ about that end?

Accepted Answer

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

Question 6

A 3.2 kg flagpole is attached to a wall at an angle of $$25^\circ$$ from the horizontal. Its center of gravity is 1.6 m from the wall. What is the magnitude of the gravitational torque about the attachment point? ($$g = 9.8 m/s$^2)$$

Accepted Answer

$$	au = (3.2 	imes 9.8 	imes 1.6) \sin 25^\circ$$

Question 7

Emma (25 kg) and Noah (22 kg) are on a seesaw. Noah sits 1.6 m from the pivot. How far from the pivot must Emma sit to balance the seesaw? (Assume seesaw mass is negligible.)

Accepted Answer

$$x_E = -\frac{22}{25} 	imes 1.6 m$$

Question 8

A solid cylinder (mass $$M$$, radius $$R) $$rotates about its central axis. What is its moment of inertia?

Accepted Answer

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

Question 9

A 2.5 kg bucket is attached to a rope wound around a 3.0 kg, 4.0-cm-radius cylinder. If the bucket is released from rest 4.0 m above the ground, which equation correctly relates the linear acceleration $$a of $$the bucket to the system's parameters?

Accepted Answer

$$a = \frac{g}{1 + \frac{M}{m}}$$

Question 10

A flywheel (solid cylinder, $$M = 540 kg$$, $$R = 0.30 m) is $$spun up by a constant torque. If the final angular speed is $$\omega_f = 1360$$ rad/s, what is the moment of inertia $$I of $$the flywheel?

Accepted Answer

$$I = \frac{1}{2} 	imes 540 	imes (0.30)^2$

Question 11

A rolling wheel of radius $$R$$ moves without slipping at speed $$v. $$What is the angular velocity $$\omega of $$the wheel?

Accepted Answer

$$\omega = \frac{v}{R}$$

Question 12

A force $$F is $$applied perpendicular to a wrench at a distance $$r$$ from the pivot. What is the maximum torque produced?

Accepted Answer

$$	au_{max} = rF$$

Question 13

A gymnast hangs horizontally from a bar. Where does the gravitational torque act?

Accepted Answer

At the gymnast's center of gravity

Question 14

Which of the following statements about the moment of inertia is correct?

Accepted Answer

It depends on both the mass and the distribution of mass relative to the axis of rotation.

Question 15

A uniform disk and a uniform hoop, both with mass $$M$$ and radius $$R$$, rotate about their centers. Which has the greater moment of inertia?

Accepted Answer

The hoop

Question 16

A force is applied at an angle to a wrench. Which component of the force produces torque?

Accepted Answer

The component perpendicular to the radial line

Question 17

A rolling object moves forward one circumference in one revolution. If the radius is $$R$$, what is the distance traveled in one revolution?

Accepted Answer

$$2\pi R$$

Question 18

A torque that tends to rotate an object in a clockwise direction is considered:

Accepted Answer

Negative

Question 19

If a net torque acts on a rigid body, what is the resulting effect according to Newton’s second law for rotation?

Accepted Answer

The body undergoes an angular acceleration

Rotational Motion: Physics with Calculus Study Notes

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