Q: A force $$F is $$applied at a distance $$r$$ from the axis at an angle $$\theta. $$Which component of the force produces torque?

The perpendicular component: $$F_\perp = F \sin \theta$$

Q: If a system is both translating and rotating, which laws must be applied?

Both $$\sum F = m a$$ and $$\sum \tau = I \alpha$$

Question 1

Which equation correctly expresses the torque $$	au$$ produced by a force $$F$$ applied at a distance $$r$$ from the axis, at an angle $$	heta$$ from the lever arm?

Accepted Answer

$$	au = F r \sin 	heta$$

Question 2

A solid cylinder of mass $$M$$ and radius $$R$$ rotates about its center. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{2} M R^2$

Question 3

You need to hold a door against a zombie pressing with a force of $$30\ \mathrm{N} at an $$angle of $$30^\circ$$ from normal at a radius of $$0.4\ \mathrm{m}. $$What minimum force must you apply at the edge of the door ($$0.8\ \mathrm{m}) at $$normal incidence to keep the door closed?

Accepted Answer

$$13.0\ \mathrm{N}$$

Question 4

A see-saw of length $$1.0\ \mathrm{m}$$ and mass $$10\ \mathrm{kg} is at an $$angle of $$30^\circ$$ with respect to the ground. The pivot is $$0.6\ \mathrm{m}$$ from one end. What mass should be added to the raised end to barely start rotation?

Accepted Answer

$$2.5\ \mathrm{kg}$$

Question 5

Which of the following is the correct form of Newton's Second Law for rotation?

Accepted Answer

$$\sum 	au = I \alpha$$

Question 6

A merry-go-round (solid cylinder, $$M = 200\ \mathrm{kg}$$, $$R = 1\ \mathrm{m}) is $$started by a $$200\ \mathrm{N}$$ force at the edge. What is the angular acceleration produced?

Accepted Answer

$$2\ \mathrm{$$rad/s^2$$}$$

Question 7

How long does it take for the merry-go-round to reach an angular speed of $$10\ \mathrm{rad/s}$$, starting from rest with the angular acceleration found above?

Accepted Answer

$$5\ \mathrm{s}$$

Question 8

A $$2\ \mathrm{kg}$$ bucket is attached to a massless string wrapped around a $$4\ \mathrm{cm}$$ diameter cylinder. If the bucket falls $$2\ \mathrm{m}$$, how does the moment of inertia of the cylinder affect the time to reach the bottom?

Accepted Answer

It increases the time, because rotational inertia resists acceleration.

Question 9

Which object has the largest moment of inertia for the same mass and radius?

Accepted Answer

A thin hoop about its center: $$I = M R^2$

Question 10

If a force is applied tangentially to a rotating object, which variable does NOT affect the torque produced?

Accepted Answer

The mass of the object

Question 11

A long uniform rod of length $$L$$ and mass $$M$$ rotates about its center. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{12} M L^2$

Question 12

Which step should you take first when solving a rotational dynamics problem?

Accepted Answer

Draw a diagram and label the forces

Question 13

A force $$F is $$applied at a distance $$r$$ from the axis at an angle $$	heta. $$Which component of the force produces torque?

Accepted Answer

The perpendicular component: $$F_\perp = F \sin 	heta$$

Question 14

If the net torque on a system is zero, what can be said about its angular acceleration?

Accepted Answer

Angular acceleration is zero: $$\alpha = 0$$

Question 15

A uniform sphere of mass $$M$$ and radius $$R$$ rotates about its center. What is its moment of inertia?

Accepted Answer

$$I = \frac{2}{5} M R^2$

Question 16

Which of the following best describes the difference between translational and rotational dynamics?

Accepted Answer

Translational dynamics uses $$\sum F = m a$$, rotational uses $$\sum 	au = I \alpha$$

Question 17

A hollow cylinder of mass $$M$$ and radius $$R$$ rotates about its center. What is its moment of inertia?

Accepted Answer

$$I = M R^2$

Question 18

If a system is both translating and rotating, which laws must be applied?

Accepted Answer

Both $$\sum F = m a$$ and $$\sum 	au = I \alpha$$

Question 19

A rectangular plate of mass $$M$$, length $$l$$, and width $$w$$ rotates about its center. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{12} M (l^2$ + w^2$)$$

Rotation II: Torque, Rotational Dynamics, and Problem Solving

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