Q: A force of $$10\ \mathrm{N} is $$applied tangentially to the rim of a wheel of radius $$0.2\ \mathrm{m}. $$What is the magnitude of the torque produced?

$$\tau = rF = 2.0\ \mathrm{N\cdot m}$$

Q: A barometer reads $$76\ \mathrm{cm} of $$mercury. What is the atmospheric pressure in Pascals? (Density of mercury $$= 13,600\ \mathrm{kg/m^3$}$$, $$g = 9.8\ \mathrm{m/s^2$})$$

$$P = \rho gh = 1.01 \times 10^5$\ \mathrm{Pa}$$

Question 1

A solid disk of mass $$M = 2.0\ \mathrm{kg}$$ and radius $$R = 0.5\ \mathrm{m}$$ rotates about its central axis. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{2}MR^2$ = 0.25\ \mathrm{kg\cdot m^2$}$$

Question 2

A wheel starts from rest and undergoes a constant angular acceleration of $$\alpha = 2.0\ \mathrm{$$rad/s^2$$}. $$What is its angular velocity after $$t = 4.0\ \mathrm{s}$$?

Accepted Answer

$$\omega = \$$omega_0$$ + \alpha t = 8.0\ \mathrm{rad/s}$$

Question 3

A uniform rod of length $$L$$ and mass $$M$$ rotates about an axis through one end perpendicular to its length. What is its moment of inertia?

Accepted Answer

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

Question 4

A force of $$10\ \mathrm{N} is $$applied tangentially to the rim of a wheel of radius $$0.2\ \mathrm{m}. $$What is the magnitude of the torque produced?

Accepted Answer

$$	au = rF = 2.0\ \mathrm{N\cdot m}$$

Question 5

A solid sphere and a solid cylinder, both of mass $$M$$ and radius $$R$$, roll without slipping down an incline. Which reaches the bottom first and why?

Accepted Answer

The sphere, because it has a smaller moment of inertia relative to its mass and radius.

Question 6

A disk rotates with constant angular velocity $$\omega. $$What is the linear speed $$v of a $$point at a distance $$r$$ from the center?

Accepted Answer

$$v = r\omega$$

Question 7

A figure skater pulls in her arms to decrease her moment of inertia by half while spinning. If her initial angular velocity is $$\$$omega_0$$, what is her final angular velocity?

Accepted Answer

$$\omega_f = 2\$$omega_0$$

Question 8

A cylinder of mass $$2.0\ \mathrm{kg}$$ and radius $$0.1\ \mathrm{m}$$ rolls without slipping at $$3.0\ \mathrm{m/s}. $$What is its total kinetic energy?

Accepted Answer

$$KE = \frac{1}{2}mv^2$ + \frac{1}{2}I\$$omega^2$$ = 12.0\ \mathrm{J}$$

Question 9

A $$5.0\ \mathrm{kg}$$ object is submerged in water and experiences a buoyant force of $$40\ \mathrm{N}. $$What is the volume of the object? (Take $$g = 10\ \mathrm{m/s^2$}$$, $$\rho_{water} = 1000\ \mathrm{kg/m^3$})$$

Accepted Answer

$$V = \frac{F_b}{\$$rho_{water}$$g} = 0.004\ \mathrm{m^3$}$$

Question 10

A hydraulic lift has a small piston of area $$0.01\ \mathrm{m^2$}$$ and a large piston of area $$0.5\ \mathrm{m^2$}. If a $$force of $$100\ \mathrm{N} is $$applied to the small piston, what is the maximum weight that can be lifted by the large piston (neglecting friction)?

Accepted Answer

$$F_{out} = F_{in} \frac{A_{out}}{A_{in}} = 5000\ \mathrm{N}$$

Question 11

A barometer reads $$76\ \mathrm{cm} of $$mercury. What is the atmospheric pressure in Pascals? (Density of mercury $$= 13,600\ \mathrm{kg/m^3$}$$, $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$P = ho gh = 1.01 	imes 10^5$\ \mathrm{Pa}$$

Question 12

A fluid flows through a pipe that narrows from a cross-sectional area of $$0.05\ \mathrm{m^2$} to$$ $$0.01\ \mathrm{m^2$}. If $$the speed in the wider section is $$2.0\ \mathrm{m/s}$$, what is the speed in the narrow section?

Accepted Answer

$$v_2$$ = $$v_1$$ \frac{$$A_1$$}{$$A_2$$} = 10.0\ \mathrm{m/s}$$

Question 13

A $$3.0\ \mathrm{kg}$$ mass is attached to a string wound around a pulley of radius $$0.2\ \mathrm{m}$$ and moment of inertia $$0.12\ \mathrm{kg\cdot m^2$}. If $$the mass is released from rest, what is the angular acceleration of the pulley? (Neglect friction)

Accepted Answer

$$\alpha = \frac{mgr}{I + mr^2$} = 4.0\ \mathrm{$$rad/s^2$$}$$

Question 14

A manometer contains water and mercury. If the difference in mercury levels is $$12\ \mathrm{cm}$$, what is the pressure difference between the two sides? (Density of mercury $$= 13,600\ \mathrm{kg/m^3$}$$, $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$\Delta P = ho_{Hg} g h = 1.6 	imes 10^4$\ \mathrm{Pa}$$

Question 15

A $$0.5\ \mathrm{kg}$$ ball is swung in a horizontal circle of radius $$1.5\ \mathrm{m} at an $$angular speed of $$4.0\ \mathrm{rad/s}. $$What is its angular momentum about the center?

Accepted Answer

$$L = m r^2$ \omega = 4.5\ \mathrm{kg\cdot m^2$/s}$$

Question 16

A $$2.0\ \mathrm{kg}$$ object is dropped into a fluid and sinks at constant velocity. What can be said about the forces acting on the object?

Accepted Answer

The sum of the buoyant force and the drag force equals the object's weight.

Question 17

A rotating platform speeds up from $$2.0\ \mathrm{rad/s} to$$ $$8.0\ \mathrm{rad/s} in$$ $$3.0\ \mathrm{s}. $$What is the average angular acceleration?

Accepted Answer

$$\alpha = \frac{\Delta \omega}{\Delta t} = 2.0\ \mathrm{$$rad/s^2$$}$$

Question 18

A pipe carries water from a reservoir at a height of $$10\ \mathrm{m}$$ above the outlet. If the water exits at atmospheric pressure and negligible speed at the top, what is the exit speed? (Neglect viscosity, $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$v = \sqrt{2gh} = 14\ \mathrm{m/s}$$

Question 19

A $$0.2\ \mathrm{kg}$$ mass is attached to a string $$0.5\ \mathrm{m}$$ from the axis and rotates at $$6.0\ \mathrm{rad/s}. $$What is the centripetal acceleration?

Accepted Answer

$$a_c = r\$$omega^2$$ = 18\ \mathrm{m/s^2$}$$

PHY 251 Exam 3: Rotational Motion, Torque, Angular Momentum, and Fluid Mechanics

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