Question 1

Which of the following is a necessary condition for an object to be in static equilibrium?

Accepted Answer

The sum of all external forces and the sum of all external torques about any axis must be zero: $$\sum F_x = 0$$, $$\sum F_y = 0$$, $$\sum 	au = 0$$

Question 2

A seesaw is balanced with a 60 N child sitting at a distance $$x$$ from the pivot and a 25 N child sitting 2.0 m from the pivot on the opposite side. What is the value of $$x$$ for equilibrium?

Accepted Answer

$$x = 0.83\ \mathrm{m}$$

Question 3

A girl weighing 200 N is sitting 3 m from the pivot of a plank. She is handed a 40 N bag of apples. To maintain balance, where should her seat be hung?

Accepted Answer

2.5 m from the pivot

Question 4

A 1.0 kg plank is held horizontally by a rope at one end and a hinge at the other. A 10 kg box is placed at a distance $$x$$ from the hinge. If the maximum tension the rope can sustain is 58 N, what is the maximum value of $$x$$ before the rope breaks? (Assume $$g = 9.8\ \mathrm{m/s^2$}$$ and the rope is at $$30^\circ$$ above the horizontal.)

Accepted Answer

$$x \approx 0.45\ \mathrm{m}$$

Question 5

A disk of moment of inertia $$I_1 = 3$$\ \mathrm{kg\ $$m^2$$}$$ rotates at $$\$$omega_1 = 4$$\ \mathrm{rad/s}$$. Another disk of I_2$ = 1\ \mathrm{kg\ m^2$} is $$dropped onto it (initially at rest). What is the final angular velocity $$\omega_f of $$the combined system?

Accepted Answer

$$\omega_f = 3\ \mathrm{rad/s}$$

Question 6

A small block of mass $$0.0250\ \mathrm{kg} is $$attached to a cord passing through a hole in a frictionless table. It revolves at $$r_1 = 0.3$$\ \mathrm{m}$$ with angular speed $$\$$omega_1 = 1.50$$\ \mathrm{rad/s}$$. The cord is pulled so the radius becomes r_2$ = 0.100\ \mathrm{m}. $$What is the new angular speed $$\$$omega_2$$?

Accepted Answer

$$\$$omega_2$$ = 13.5\ \mathrm{rad/s}$$

Question 7

For the block in the previous question, what is the change in kinetic energy after the radius is reduced from $$0.3\ \mathrm{m} to$$ $$0.1\ \mathrm{m}$$?

Accepted Answer

$$\Delta K = 1.69\ \mathrm{J}$$

Question 8

A system consists of two masses, $$3m$$ and $$m$$, connected by a rope over a pulley of mass $$M$$ and radius $$R. If $$the pulley is a uniform disk, which equation correctly relates the acceleration $$a of $$the masses to the system parameters?

Accepted Answer

$$a = \dfrac{2mg}{M/2 + 4m}$$

Question 9

Which of the following statements about torque is correct?

Accepted Answer

Torque depends on both the magnitude of the force and the perpendicular distance from the axis of rotation: $$	au = rF\sin	heta$$

Question 10

A horizontal beam is supported at its ends. A 50 kg person stands 2 m from the left end of the 4 m beam (mass 20 kg). What is the upward force at the left support?

Accepted Answer

$$F_{left} = 490\ \mathrm{N}$$

Question 11

A uniform plank of length $$L$$ and mass $$M is $$supported horizontally at its ends. A person of mass $$m$$ stands at a distance $$d$$ from the left end. What is the reaction force at the right support?

Accepted Answer

$$F_{right} = \dfrac{m g d + M g \frac{L}{2}}{L}$$

Question 12

A turntable of moment of inertia $$I_1$$ rotates at angular velocity $$\$$omega_1$$. A second disk of moment of inertia I_2$ is dropped onto it and sticks. Which principle is used to find the final angular velocity?

Accepted Answer

Conservation of angular momentum: $$I_1$$ \$$omega_1$$ = ($$I_1$$ + $$I_2$$) \omega_f$$

Question 13

A block is attached to a cord passing through a hole in a frictionless table. If the radius of revolution is halved, what happens to the angular speed?

Accepted Answer

It increases by a factor of 4

Question 14

Which of the following best describes the effect of choosing the axis of rotation when calculating torques for equilibrium?

Accepted Answer

Choosing the axis can simplify calculations, but the condition $$\sum 	au = 0$$ must hold for any axis

Question 15

A uniform beam of length $$L$$ and mass $$M is $$supported horizontally at its ends. If a person of mass $$m$$ stands at the very end of the beam, what is the reaction force at the opposite support?

Accepted Answer

$$F_{opposite} = \dfrac{M g}{2}$$

Question 16

A plank is balanced on a pivot. If a child of weight $$W$$ sits at a distance $$d$$ from the pivot, what is the torque produced by the child about the pivot?

Accepted Answer

$$	au = W d$$

Question 17

A uniform disk of mass $$M$$ and radius $$R is $$used as a pulley. If a mass $$m$$ hangs from a rope wrapped around the disk, what is the acceleration of the mass?

Accepted Answer

$$a = \dfrac{g}{1 + \frac{M}{2m}}$$

Question 18

A beam is supported at its ends and loaded with a point mass at its center. What is the reaction force at each support?

Accepted Answer

$$F_{support} = \dfrac{1}{2} m g$$

Equilibrium, Torque, and Rotational Dynamics Study Notes

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