Q: A mechanic uses a wrench to loosen a bolt. If the mechanic applies a force of 200 N at the end of a 0.3 m long wrench at an angle of 90° to the wrench, what is the magnitude of the torque produced about the bolt?

$$\tau = 200 \times 0.3 \times \sin(90^\circ) = 60\ \mathrm{N\cdot m}$$

Q: A seesaw is balanced with a 40 kg child sitting 2 m from the pivot and a 60 kg child sitting on the other side. How far from the pivot must the 60 kg child sit to maintain equilibrium? (Assume both children sit perpendicular to the plank.)

$$40 \times 2 = 60 \times x \Rightarrow x = \frac{80}{60} = 1.33\ \mathrm{m}$$

Q: A uniform beam of length 4 m and weight 200 N is attached to a wall at one end and supported by a cable at the other end. If a person weighing 600 N stands 1.5 m from the wall, what is the total torque about the wall due to the person and the beam's weight?

$$\tau_{\text{beam}} = 2 \times 200 = 400\ \mathrm{N\cdot m}$$, $$\tau_{\text{person}} = 1.5 \times 600 = 900\ \mathrm{N\cdot m}$$, Total = $$400 + 900 = 1300\ \mathrm{N\cdot m}$$

Q: A door of mass 20 kg and width 1 m is pushed with a force of 50 N at the edge perpendicular to the door. What is the angular acceleration of the door? (Moment of inertia for a rod about one end: $$I = \frac{1}{3}ML^2$)

$$I = \frac{1}{3} \times 20 \times 1^2$ = 6.67\ \mathrm{kg\cdot m^2$}$$, $$\tau = 50 \times 1 = 50\ \mathrm{N\cdot m}$$, $$\alpha = \frac{50}{6.67} = 7.5\ \mathrm{$$rad/s^2$$}$$

Question 1

A mechanic uses a wrench to loosen a bolt. If the mechanic applies a force of 200 N at the end of a 0.3 m long wrench at an angle of 90° to the wrench, what is the magnitude of the torque produced about the bolt?

Accepted Answer

$$	au = 200 	imes 0.3 	imes \sin(90^\circ) = 60\ \mathrm{N\cdot m}$$

Question 2

Which of the following statements about rotational equilibrium is correct?

Accepted Answer

An object is in rotational equilibrium if the net external torque acting on it is zero.

Question 3

A seesaw is balanced with a 40 kg child sitting 2 m from the pivot and a 60 kg child sitting on the other side. How far from the pivot must the 60 kg child sit to maintain equilibrium? (Assume both children sit perpendicular to the plank.)

Accepted Answer

$$40 	imes 2 = 60 	imes x \Rightarrow x = \frac{80}{60} = 1.33\ \mathrm{m}$$

Question 4

A uniform beam of length 4 m and weight 200 N is attached to a wall at one end and supported by a cable at the other end. If a person weighing 600 N stands 1.5 m from the wall, what is the total torque about the wall due to the person and the beam's weight?

Accepted Answer

$$	au_{	ext{beam}} = 2 	imes 200 = 400\ \mathrm{N\cdot m}$$, $$	au_{	ext{person}} = 1.5 	imes 600 = 900\ \mathrm{N\cdot m}$$, Total = $$400 + 900 = 1300\ \mathrm{N\cdot m}$$

Question 5

Which of the following best describes the center of gravity of a uniform object?

Accepted Answer

It is located at the geometric center of the object.

Question 6

A gymnast spins faster when she pulls her arms in close to her body. Which principle explains this phenomenon?

Accepted Answer

Conservation of angular momentum: $$L = I \omega$$ remains constant, so decreasing $$I$$ increases $$\omega.$$

Question 7

A solid sphere of mass $$M$$ and radius $$R$$ rolls down a frictionless incline of height $$h. $$What is its final linear speed at the bottom if it rolls without slipping?

Accepted Answer

$$v = \sqrt{\frac{10}{7}gh}$$

Question 8

A ring and a solid ball of equal mass and radius roll down the same incline. Which reaches the bottom first and why?

Accepted Answer

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

Question 9

A person sits on a rotating stool holding weights. If the person pulls the weights inward, what happens to the angular speed of the system?

Accepted Answer

It increases because the moment of inertia decreases and angular momentum is conserved.

Question 10

Which equation correctly relates torque ($$	au$$), moment of inertia ($$I$$), and angular acceleration ($$\alpha$$)?

Accepted Answer

$$	au = I \alpha$$

Question 11

A door of mass 20 kg and width 1 m is pushed with a force of 50 N at the edge perpendicular to the door. What is the angular acceleration of the door? (Moment of inertia for a rod about one end: $$I = \frac{1}{3}ML^2$)

Accepted Answer

$$I = \frac{1}{3} 	imes 20 	imes 1^2$ = 6.67\ \mathrm{kg\cdot m^2$}$$, $$	au = 50 	imes 1 = 50\ \mathrm{N\cdot m}$$, $$\alpha = \frac{50}{6.67} = 7.5\ \mathrm{$$rad/s^2$$}$$

Question 12

Which of the following is NOT a condition for mechanical equilibrium?

Accepted Answer

The net external angular velocity must be zero.

Question 13

A baton consists of four spheres (each of mass $$m) $$attached to the ends of two light rods of length $$l$$ crossing at the center. What is the moment of inertia about the center?

Accepted Answer

$$I = m l^2$ + m l^2$ + m l^2$ + m l^2$ = 4 m l^2$

Question 14

A revolving door is pushed by two people at different radii and forces. If one pushes with 625 N at 1.2 m (CW) and the other with 850 N at 0.8 m (CCW), what is the net torque?

Accepted Answer

$$	au_{	ext{CW}} = -1.2 	imes 625 = -750\ \mathrm{N\cdot m}$$, $$	au_{	ext{CCW}} = 0.8 	imes 850 = 680\ \mathrm{N\cdot m}$$, Net = $$-750 + 680 = -70\ \mathrm{N\cdot m}$$

Question 15

A ball rolls down a slope without slipping. If the surface were frictionless and the ball slipped instead, what would its final linear speed be at the bottom?

Accepted Answer

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

Question 16

Which of the following is true about the work done when a spinning person pulls weights inward on a rotating stool?

Accepted Answer

The work equals the increase in rotational kinetic energy: $$W = \frac{1}{2}I_f \omega_f^2$ - \frac{1}{2}I_i \omega_i^2$

Question 17

A person holds a 50 N ball in their hand with their forearm horizontal. If the biceps attaches 3 cm from the elbow and the ball is 35 cm from the elbow, what is the force exerted by the biceps?

Accepted Answer

$$F = \frac{0.35 	imes 50}{0.03} = 583\ \mathrm{N}$$

Question 18

If a merry-go-round is rotating and a person walks toward the center, what happens to the angular speed of the merry-go-round (assuming no external torque)?

Accepted Answer

It increases because the moment of inertia decreases and angular momentum is conserved.

Rotational Equilibrium and Dynamics: Torque, Lever Arm, and Applications

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