Q: A 2 kg object moves in a horizontal circle of radius 0.5 m at a constant speed of 4 m/s. What is the magnitude of the net force acting on the object?

$$F = m a_c = m \frac{v^2$}{r} = 2 \times \frac{16}{0.5} = 64\ \mathrm{N}$$

Q: A satellite orbits Earth in a circular orbit of radius $$7.0 \times 10^6$ m. If the mass of Earth is 6.0$$ \times $$10^{24} kg$$, what is the speed of the satellite?

$$v = \sqrt{\frac{GM}{r}} = \sqrt{\frac{6.67 \times 10$$^{-11}$$ \times 6.0 \times 10$$^{24}$$}{7.0 \times 10^6$}} \approx 7.56 \times 10^3$\ \mathrm{m/s}$$

Q: A 0.5 kg mass is attached to a spring with a spring constant $$k = 200\ \mathrm{N/m}. If $$the spring is stretched by 0.1 m, what is the force exerted by the spring?

$$F = -k \Delta x = -200 \times 0.1 = -20\ \mathrm{N}$$

Q: A solid disk of mass 2 kg and radius 0.3 m rotates about its center. What is its moment of inertia?

$$I = \frac{1}{2} M R^2$ = \frac{1}{2} \times 2 \times (0.3)^2$ = 0.09\ \mathrm{kg\cdot m^2$}$$

Q: A force of 10 N is applied perpendicular to a wrench at a distance of 0.2 m from the pivot. What is the torque produced?

$$\tau = r F \sin \theta = 0.2 \times 10 \times \sin 90^\circ = 2\ \mathrm{N\cdot m}$$

Q: A wheel starts from rest and has a constant angular acceleration of $$2\ \mathrm{$$rad/s^2$$}. $$What is its angular velocity after 5 seconds?

$$\omega = \$$omega_0$$ + \alpha t = 0 + 2 \times 5 = 10\ \mathrm{rad/s}$$

Q: A uniform rod of length 2 m and mass 3 kg rotates about an axis through its center. What is its moment of inertia?

$$I = \frac{1}{12} M L^2$ = \frac{1}{12} \times 3 \times (2)^2$ = 1\ \mathrm{kg\cdot m^2$}$$

Q: A 5 kg object is attached to a string and swung in a horizontal circle of radius 2 m at a speed of 6 m/s. What is the tension in the string?

$$T = m \frac{v^2$}{r} = 5 \times \frac{36}{2} = 90\ \mathrm{N}$$

Question 1

A 2 kg object moves in a horizontal circle of radius 0.5 m at a constant speed of 4 m/s. What is the magnitude of the net force acting on the object?

Accepted Answer

$$F = m a_c = m \frac{v^2$}{r} = 2 	imes \frac{16}{0.5} = 64\ \mathrm{N}$$

Question 2

A satellite orbits Earth in a circular orbit of radius $$7.0 	imes 10^6$ m. If the mass of Earth is 6.0$$ 	imes $$10^{24} kg$$, what is the speed of the satellite?

Accepted Answer

$$v = \sqrt{\frac{GM}{r}} = \sqrt{\frac{6.67 	imes 10$$^{-11}$$ 	imes 6.0 	imes 10$$^{24}$$}{7.0 	imes 10^6$}} \approx 7.56 	imes 10^3$\ \mathrm{m/s}$$

Question 3

Which of the following statements about centripetal acceleration is correct?

Accepted Answer

It is always directed toward the center of the circle.

Question 4

A 0.5 kg mass is attached to a spring with a spring constant $$k = 200\ \mathrm{N/m}. If $$the spring is stretched by 0.1 m, what is the force exerted by the spring?

Accepted Answer

$$F = -k \Delta x = -200 	imes 0.1 = -20\ \mathrm{N}$$

Question 5

A solid disk of mass 2 kg and radius 0.3 m rotates about its center. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{2} M R^2$ = \frac{1}{2} 	imes 2 	imes (0.3)^2$ = 0.09\ \mathrm{kg\cdot m^2$}$$

Question 6

A force of 10 N is applied perpendicular to a wrench at a distance of 0.2 m from the pivot. What is the torque produced?

Accepted Answer

$$	au = r F \sin 	heta = 0.2 	imes 10 	imes \sin 90^\circ = 2\ \mathrm{N\cdot m}$$

Question 7

A wheel starts from rest and has a constant angular acceleration of $$2\ \mathrm{$$rad/s^2$$}. $$What is its angular velocity after 5 seconds?

Accepted Answer

$$\omega = \$$omega_0$$ + \alpha t = 0 + 2 	imes 5 = 10\ \mathrm{rad/s}$$

Question 8

A uniform rod of length 2 m and mass 3 kg rotates about an axis through its center. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{12} M L^2$ = \frac{1}{12} 	imes 3 	imes (2)^2$ = 1\ \mathrm{kg\cdot m^2$}$$

Question 9

A 5 kg object is attached to a string and swung in a horizontal circle of radius 2 m at a speed of 6 m/s. What is the tension in the string?

Accepted Answer

$$T = m \frac{v^2$}{r} = 5 	imes \frac{36}{2} = 90\ \mathrm{N}$$

Question 10

A 0.2 kg mass is attached to a spring and oscillates with a period of 0.5 s. What is the spring constant $$k$$?

Accepted Answer

$$T = 2\pi \sqrt{\frac{m}{k}} \implies k = \frac{4\pi^2$ m}{T^2$} = \frac{4 	imes \pi^2$ 	imes 0.2}{0.25} \approx 31.6\ \mathrm{N/m}$$

Question 11

A 10 kg beam is 4 m long and is supported at its ends. A 20 kg mass is placed 1 m from the left end. What is the force on the left support?

Accepted Answer

Let $$F_L$$ and $$F_R be $$the forces at the left and right supports. Taking torques about the left end: $$F_R 	imes 4 = 10 	imes 9.8 	imes 2 + 20 	imes 9.8 	imes 1$$; $$F_R = (196 + 196)/4 = 98\ \mathrm{N}$$; $$F_L + F_R = (10+20)	imes 9.8 = 294\ \mathrm{N}$$; $$F_L = 294 - 98 = 196\ \mathrm{N}$$

Question 12

Which of the following is true for an object in static equilibrium?

Accepted Answer

$$\sum F = 0$$ and $$\sum 	au = 0$$

Question 13

A hoop of mass 1 kg and radius 0.5 m rolls without slipping at 2 m/s. What is its total kinetic energy?

Accepted Answer

Translational: $$K_t = \frac{1}{2} m v^2$ = 0.5 	imes 1 	imes 4 = 2\ \mathrm{J}$$; Rotational: $$K_r = \frac{1}{2} I \$$omega^2$$, I$$ = $$mR^2 = 0.25$, $$\omega = v/R = 4$$; K_r = 0.5$$ 	imes 0.25 	imes 16 = 2\ \mathrm{J}$$; Total = 4$$\ \mathrm{J}$$

Question 14

A 3 kg object is attached to a string and swung in a vertical circle of radius 1.5 m. What is the minimum speed at the top of the circle to keep the string taut?

Accepted Answer

At the top, $$T + mg = m v^2$ / r$$, for minimum $$T=0$$, $$v = \sqrt{g r} = \sqrt{9.8 	imes 1.5} \approx 3.83\ \mathrm{m/s}$$

Question 15

A disk rotates from rest with a constant angular acceleration of $$3\ \mathrm{$$rad/s^2$$}. $$How many radians does it rotate in 4 seconds?

Accepted Answer

$$	heta = \$$omega_0$$ t + \frac{1}{2} \alpha t^2$ = 0 + 0.5 	imes 3 	imes 16 = 24\ \mathrm{rad}$$

Question 16

A 2 kg mass is attached to a string and whirled in a horizontal circle of radius 1 m at a frequency of 2 revolutions per second. What is the tension in the string?

Accepted Answer

Frequency $$f = 2\ \mathrm{Hz}$$, $$v = 2 \pi r f = 2 \pi 	imes 1 	imes 2 = 4\pi\ \mathrm{m/s}$$; $$T = m v^2$ / r = 2 	imes (4\pi)^2$ / 1 = 2 	imes 16 \pi^2$ = 32 \pi^2$ \approx 316\ \mathrm{N}$$

Question 17

A uniform beam of length 3 m and mass 12 kg is supported horizontally at its ends. A 24 kg mass is hung 1 m from the left end. What is the force on the right support?

Accepted Answer

Let $$F_L$$ and $$F_R be $$the left and right supports. Take torques about the left end: $$F_R 	imes 3 = 12 	imes 9.8 	imes 1.5 + 24 	imes 9.8 	imes 1$$; $$F_R = (176.4 + 235.2)/3 = 137.2\ \mathrm{N}$$

Question 18

A solid sphere of mass 4 kg and radius 0.2 m rolls without slipping down an incline. What is its acceleration?

Accepted Answer

For a solid sphere, $$a = \frac{5}{7} g \sin 	heta ($$from $$a = \frac{g \sin 	heta}{1 + I/(mR^2$)}$$, $$I = 2/5 mR^2$)

Physics Review: Circular Motion, Rotational Motion, and Equilibrium

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