Q: A 5 kg box is at rest on a horizontal surface. The coefficient of static friction between the box and the surface is 0.4. What is the minimum horizontal force required to start moving the box?

$$F = \mu_s m g = 0.4 \times 5 \times 9.8 = 19.6\ \mathrm{N}$$

Q: A 2 kg mass is attached to a string and swung in a horizontal circle of radius 1.5 m at a constant speed of 4 m/s. What is the tension in the string?

$$T = m a_c = m \frac{v^2$}{r} = 2 \times \frac{16}{1.5} = 21.3\ \mathrm{N}$$

Q: A car of mass 1200 kg rounds a flat curve of radius 50 m at a speed of 18 m/s. What is the minimum coefficient of static friction required to prevent the car from skidding?

$$\mu_s = \frac{v^2$}{r g} = \frac{324}{50 \times 9.8} = 0.66$$

Q: A 10 kg block is pulled up a rough incline (angle $$30^\circ) $$with a force parallel to the incline. The coefficient of kinetic friction is 0.2. What is the magnitude of the frictional force?

$$f_k = \mu_k N = 0.2 \times 10 \times 9.8 \times \cos 30^\circ = 0.2 \times 10 \times 9.8 \times 0.866 = 16.97\ \mathrm{N}$$

Q: A 3 kg object is suspended from a spring with spring constant $$k = 150\ \mathrm{N/m}. $$How much does the spring stretch from its equilibrium position?

$$\Delta x = \frac{m g}{k} = \frac{3 \times 9.8}{150} = 0.196\ \mathrm{m}$$

Q: A 1.5 kg object is attached to a rope and swung in a vertical circle of radius 2 m. At the lowest point, the speed is 6 m/s. What is the tension in the rope at this point?

$$T = m g + m \frac{v^2$}{r} = 1.5 \times 9.8 + 1.5 \times \frac{36}{2} = 14.7 + 27 = 41.7\ \mathrm{N}$$

Q: A 20 kg box is suspended by two ropes, each making a $$45^\circ$$ angle with the ceiling. What is the tension in each rope?

$$T = \frac{m g}{2 \sin 45^\circ} = \frac{20 \times 9.8}{2 \times 0.707} = 138.6\ \mathrm{N}$$

Q: A 4 kg block is connected by a light string over a frictionless pulley to a 6 kg block hanging vertically. The 4 kg block rests on a horizontal surface with a coefficient of kinetic friction $$\mu_k = 0.1. $$What is the acceleration of the system?

$$a = \frac{m_2$ g - f_k}{m_1$ + m_2$} = \frac{6 \times 9.8 - 0.1 \times 4 \times 9.8}{4 + 6} = \frac{58.8 - 3.92}{10} = 5.49\ \mathrm{m/s^2$}$$

Q: A 70 kg person stands on a scale in an elevator accelerating upward at $$2\ \mathrm{m/s^2$}. $$What does the scale read?

$$N = m(g + a) = 70 \times (9.8 + 2) = 826\ \mathrm{N}$$

Q: A 0.5 kg ball is attached to a string and swung in a horizontal circle of radius 0.8 m. If the string can withstand a maximum tension of 20 N, what is the maximum speed the ball can have before the string breaks?

$$v_{max} = \sqrt{\frac{T_{max} r}{m}} = \sqrt{\frac{20 \times 0.8}{0.5}} = \sqrt{32} = 5.66\ \mathrm{m/s}$$

Question 1

A 5 kg box is at rest on a horizontal surface. The coefficient of static friction between the box and the surface is 0.4. What is the minimum horizontal force required to start moving the box?

Accepted Answer

$$F = \mu_s m g = 0.4 	imes 5 	imes 9.8 = 19.6\ \mathrm{N}$$

Question 2

A 2 kg mass is attached to a string and swung in a horizontal circle of radius 1.5 m at a constant speed of 4 m/s. What is the tension in the string?

Accepted Answer

$$T = m a_c = m \frac{v^2$}{r} = 2 	imes \frac{16}{1.5} = 21.3\ \mathrm{N}$$

Question 3

A car of mass 1200 kg rounds a flat curve of radius 50 m at a speed of 18 m/s. What is the minimum coefficient of static friction required to prevent the car from skidding?

Accepted Answer

$$\mu_s = \frac{v^2$}{r g} = \frac{324}{50 	imes 9.8} = 0.66$$

Question 4

A 10 kg block is pulled up a rough incline (angle $$30^\circ) $$with a force parallel to the incline. The coefficient of kinetic friction is 0.2. What is the magnitude of the frictional force?

Accepted Answer

$$f_k = \mu_k N = 0.2 	imes 10 	imes 9.8 	imes \cos 30^\circ = 0.2 	imes 10 	imes 9.8 	imes 0.866 = 16.97\ \mathrm{N}$$

Question 5

A 3 kg object is suspended from a spring with spring constant $$k = 150\ \mathrm{N/m}. $$How much does the spring stretch from its equilibrium position?

Accepted Answer

$$\Delta x = \frac{m g}{k} = \frac{3 	imes 9.8}{150} = 0.196\ \mathrm{m}$$

Question 6

A 1.5 kg object is attached to a rope and swung in a vertical circle of radius 2 m. At the lowest point, the speed is 6 m/s. What is the tension in the rope at this point?

Accepted Answer

$$T = m g + m \frac{v^2$}{r} = 1.5 	imes 9.8 + 1.5 	imes \frac{36}{2} = 14.7 + 27 = 41.7\ \mathrm{N}$$

Question 7

A 20 kg box is suspended by two ropes, each making a $$45^\circ$$ angle with the ceiling. What is the tension in each rope?

Accepted Answer

$$T = \frac{m g}{2 \sin 45^\circ} = \frac{20 	imes 9.8}{2 	imes 0.707} = 138.6\ \mathrm{N}$$

Question 8

A 4 kg block is connected by a light string over a frictionless pulley to a 6 kg block hanging vertically. The 4 kg block rests on a horizontal surface with a coefficient of kinetic friction $$\mu_k = 0.1. $$What is the acceleration of the system?

Accepted Answer

$$a = \frac{m_2$ g - f_k}{m_1$ + m_2$} = \frac{6 	imes 9.8 - 0.1 	imes 4 	imes 9.8}{4 + 6} = \frac{58.8 - 3.92}{10} = 5.49\ \mathrm{m/s^2$}$$

Question 9

A 70 kg person stands on a scale in an elevator accelerating upward at $$2\ \mathrm{m/s^2$}. $$What does the scale read?

Accepted Answer

$$N = m(g + a) = 70 	imes (9.8 + 2) = 826\ \mathrm{N}$$

Question 10

A 0.5 kg ball is attached to a string and swung in a horizontal circle of radius 0.8 m. If the string can withstand a maximum tension of 20 N, what is the maximum speed the ball can have before the string breaks?

Accepted Answer

$$v_{max} = \sqrt{\frac{T_{max} r}{m}} = \sqrt{\frac{20 	imes 0.8}{0.5}} = \sqrt{32} = 5.66\ \mathrm{m/s}$$

Question 11

A 2 kg block is pressed against a vertical wall by a horizontal force of 50 N. The coefficient of static friction between the block and the wall is 0.3. What is the maximum weight of the block that can be supported without slipping?

Accepted Answer

$$f_{s,max} = \mu_s N = 0.3 	imes 50 = 15\ \mathrm{N}$$

Question 12

A 1 kg puck slides on a frictionless horizontal surface and is attached to a string that passes through a hole in the center of the table. If the puck moves in a circle of radius 0.5 m at a speed of 3 m/s, what is the force exerted by the string?

Accepted Answer

$$F = m \frac{v^2$}{r} = 1 	imes \frac{9}{0.5} = 18\ \mathrm{N}$$

Question 13

A 5 kg block is at rest on a horizontal table. A horizontal force of 15 N is applied, but the block does not move. What can you conclude about the coefficient of static friction?

Accepted Answer

$$\mu_s \geq \frac{F}{m g} = \frac{15}{5 	imes 9.8} = 0.31$$

Question 14

A 0.2 kg ball is attached to a spring (k = 40 N/m) and whirled in a horizontal circle of radius 0.5 m on a frictionless table. What is the speed of the ball if the spring stretches by 0.1 m?

Accepted Answer

$$F_{spring} = k x = 40 	imes 0.1 = 4\ \mathrm{N}$$;\quad F_{spring} = m \frac{$$v^2$$}{r} \implies v = \sqrt{\frac{F_{spring} r}{m}} = \sqrt{\frac{4 	imes 0.5}{0.2}} = \sqrt{10} = 3.16\ \mathrm{m/s}$

Question 15

A 10 kg crate is pulled at constant speed up a ramp inclined at $$25^\circ to $$the horizontal. If the coefficient of kinetic friction is 0.15, what is the magnitude of the pulling force parallel to the ramp?

Accepted Answer

$$F = m g (\sin 25^\circ + \mu_k \cos 25^\circ) = 10 	imes 9.8 (0.423 + 0.15 	imes 0.906) = 98 (0.423 + 0.136) = 98 	imes 0.559 = 54.8\ \mathrm{N}$$

Question 16

A 0.8 kg object moves in a vertical circle of radius 0.6 m. What is the minimum speed at the top of the circle so that the object just maintains contact with the circle?

Accepted Answer

$$v_{min} = \sqrt{g r} = \sqrt{9.8 	imes 0.6} = 2.43\ \mathrm{m/s}$$

Question 17

A 12 kg box is pushed across a horizontal floor with a force of 40 N. The coefficient of kinetic friction is 0.25. What is the acceleration of the box?

Accepted Answer

$$f_k = \mu_k m g = 0.25 	imes 12 	imes 9.8 = 29.4\ \mathrm{N}$$;\quad a = \frac{F - f_k}{m} = \frac{40 - 29.4}{12} = 0.88\ \mathrm{$$m/s^2$$}$

Question 18

A 0.3 kg mass hangs from a spring and oscillates vertically. If the spring stretches 0.04 m under the mass, what is the spring constant?

Accepted Answer

$$k = \frac{m g}{\Delta x} = \frac{0.3 	imes 9.8}{0.04} = 73.5\ \mathrm{N/m}$$

Question 19

A 2 kg block is attached to a spring (k = 100 N/m) and placed on a frictionless table. The block is pulled 0.1 m from equilibrium and released. What is the maximum force exerted by the spring?

Accepted Answer

$$F_{max} = k x_{max} = 100 	imes 0.1 = 10\ \mathrm{N}$$

Study Guide: Forces, Newton's Laws, and Circular Motion (Chapters 5-8)

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