Q: A 0.15 kg baseball is pitched at 40 m/s and is hit straight back at 60 m/s by a bat. If the contact time between the bat and the ball is 0.005 s, what is the magnitude of the average force exerted by the bat on the ball?

$$F = \frac{\Delta p}{\Delta t} = \frac{0.15 \times (60 - (-40))}{0.005} = 3000\ \mathrm{N}$$

Q: A wheel accelerates from rest at a constant angular acceleration of $$2\ \mathrm{$$rad/s^2$$}. $$How many radians does it rotate in 5 seconds?

$$\theta = \$$omega_0$$ t + \frac{1}{2} \alpha t^2$ = 0 + \frac{1}{2} \times 2 \times 25 = 25\ \mathrm{rad}$$

Q: A force of 10 N is applied tangentially to the rim of a wheel of radius 0.5 m. What is the torque produced?

$$\tau = r F = 0.5 \times 10 = 5\ \mathrm{N \cdot m}$$

Q: A cylindrical container (radius 0.1 m, height 0.5 m) is filled with oil (density $$900\ \mathrm{kg/m^3$}). $$What is the pressure at the bottom due to the oil?

$$P = \rho g h = 900 \times 9.8 \times 0.5 = 4410\ \mathrm{Pa}$$

Q: A 0.5 kg ball is dropped from a height of 2 m and bounces straight up to a height of 1.5 m. What is the magnitude of the impulse delivered by the floor?

$$J = m(v_{up} + v_{down}) = 0.5 \times (\sqrt{2 \times 9.8 \times 1.5} + \sqrt{2 \times 9.8 \times 2}) \approx 0.5 \times (5.42 + 6.26) = 5.84\ \mathrm{kg \cdot m/s}$$

Question 1

A 0.15 kg baseball is pitched at 40 m/s and is hit straight back at 60 m/s by a bat. If the contact time between the bat and the ball is 0.005 s, what is the magnitude of the average force exerted by the bat on the ball?

Accepted Answer

$$F = \frac{\Delta p}{\Delta t} = \frac{0.15 	imes (60 - (-40))}{0.005} = 3000\ \mathrm{N}$$

Question 2

A 2 kg cart moving at 3 m/s collides elastically with a stationary 1 kg cart. What is the velocity of the 2 kg cart after the collision?

Accepted Answer

$$v_{1f} = \frac{m_1$ - m_2$}{m_1$ + m_2$} v_{1i} = \frac{2 - 1}{2 + 1} 	imes 3 = 1\ \mathrm{m/s}$$

Question 3

Which of the following best describes the center of mass of a system?

Accepted Answer

It is the point where the total mass of the system can be considered to be concentrated for translational motion.

Question 4

A solid disk (mass $$M$$, radius $$R) $$rotates about its center with angular velocity $$\omega. $$What is its rotational kinetic energy?

Accepted Answer

$$K = \frac{1}{2} I \$$omega^2$$ = \frac{1}{2} \left(\frac{1}{2} M R^2$\right) \$$omega^2$$ = \frac{1}{4} M R^2$ \$$omega^2$$

Question 5

A wheel accelerates from rest at a constant angular acceleration of $$2\ \mathrm{$$rad/s^2$$}. $$How many radians does it rotate in 5 seconds?

Accepted Answer

$$	heta = \$$omega_0$$ t + \frac{1}{2} \alpha t^2$ = 0 + \frac{1}{2} 	imes 2 	imes 25 = 25\ \mathrm{rad}$$

Question 6

A force of 10 N is applied tangentially to the rim of a wheel of radius 0.5 m. What is the torque produced?

Accepted Answer

$$	au = r F = 0.5 	imes 10 = 5\ \mathrm{N \cdot m}$$

Question 7

A uniform beam of length 4 m and mass 20 kg is supported at its center. A 40 kg mass is hung 1 m from the left end. Where should a 60 kg mass be hung from the right end to balance the beam?

Accepted Answer

1.5 m from the right end

Question 8

A figure skater pulls in her arms to decrease her moment of inertia to half its original value. If her initial angular velocity was $$\$$omega_0$$, what is her final angular velocity?

Accepted Answer

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

Question 9

A cylindrical container (radius 0.1 m, height 0.5 m) is filled with oil (density $$900\ \mathrm{kg/m^3$}). $$What is the pressure at the bottom due to the oil?

Accepted Answer

$$P = ho g h = 900 	imes 9.8 	imes 0.5 = 4410\ \mathrm{Pa}$$

Question 10

A block of wood (density $$600\ \mathrm{kg/m^3$}) $$floats in water (density $$1000\ \mathrm{kg/m^3$}). $$What fraction of the block's volume is submerged?

Accepted Answer

$$\frac{V_{sub}}{V_{total}} = \frac{\rho_{block}}{\rho_{water}} = \frac{600}{1000} = 0.6$$

Question 11

A pipe narrows from a diameter of 10 cm to 5 cm. If water enters the wide end at 2 m/s, what is the speed at the narrow end?

Accepted Answer

$$A_1$$ $$v_1$$ = $$A_2$$ $$v_2$$ \Rightarrow $$v_2$$ = $$v_1$$ \left(\frac{$$d_1$$}{$$d_2$$}\$$right)^2 = 2$$ 	imes $$(2)^2 = 8$$\ \mathrm{m/s}$$

Question 12

A 0.5 kg ball is dropped from a height of 2 m and bounces straight up to a height of 1.5 m. What is the magnitude of the impulse delivered by the floor?

Accepted Answer

$$J = m(v_{up} + v_{down}) = 0.5 	imes (\sqrt{2 	imes 9.8 	imes 1.5} + \sqrt{2 	imes 9.8 	imes 2}) \approx 0.5 	imes (5.42 + 6.26) = 5.84\ \mathrm{kg \cdot m/s}$$

Question 13

A 3 kg object moves in a circle of radius 2 m at a constant speed of 4 m/s. What is its angular momentum about the center?

Accepted Answer

$$L = m v r = 3 	imes 4 	imes 2 = 24\ \mathrm{kg \cdot m^2$/s}$$

Question 14

A 5 kg object is acted on by a net torque of $$10\ \mathrm{N \cdot m}$$ about a fixed axis. If its moment of inertia is $$2\ \mathrm{kg \cdot m^2$}$$, what is its angular acceleration?

Accepted Answer

$$\alpha = \frac{	au}{I} = \frac{10}{2} = 5\ \mathrm{$$rad/s^2$$}$$

Question 15

A diver of mass 70 kg jumps off a diving board 10 m above the water. What is the diver's speed just before hitting the water (ignore air resistance)?

Accepted Answer

$$v = \sqrt{2 g h} = \sqrt{2 	imes 9.8 	imes 10} = 14\ \mathrm{m/s}$$

Question 16

A 2 kg object floats in a fluid with 75% of its volume submerged. What is the density of the fluid?

Accepted Answer

$$\rho_{fluid} = \frac{\rho_{object}}{0.75} = \frac{2000}{0.75} = 2667\ \mathrm{kg/m^3$}$$

Question 17

A 1.5 kg hoop (radius 0.4 m) rolls without slipping at 3 m/s. What is its total kinetic energy?

Accepted Answer

$$K = \frac{1}{2} m v^2$ + \frac{1}{2} I \$$omega^2$$ = \frac{1}{2} 	imes 1.5 	imes 9 + \frac{1}{2} 	imes (1.5 	imes 0.16) 	imes (7.5)^2$ = 6.75 + 6.75 = 13.5\ \mathrm{J}$$

Question 18

A 0.2 kg object is attached to a string and swung in a horizontal circle of radius 0.5 m at 4 rad/s. What is the tension in the string?

Accepted Answer

$$T = m r \$$omega^2$$ = 0.2 	imes 0.5 	imes 16 = 1.6\ \mathrm{N}$$

Question 19

A 10 kg box rests on a frictionless table. A force of 20 N is applied at an angle of 30° above the horizontal. What is the horizontal acceleration of the box?

Accepted Answer

$$a = \frac{F \cos 	heta}{m} = \frac{20 	imes 0.866}{10} = 1.73\ \mathrm{m/s^2$}$$

Exam #3 Study Guide: Linear Momentum, Rotational Motion, and Fluids

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