Q: A force of 35.0 N is required to start a 6.0-kg box moving across a horizontal concrete floor. What is the coefficient of static friction between the box and the floor?

$$\mu_s = \frac{F}{mg} = \frac{35.0}{6.0 \times 9.8} = 0.60$$

Q: A 6.0-kg box is pushed with a constant force of 35.0 N and accelerates at $$0.60\ \mathrm{m/s^2$}. $$What is the coefficient of kinetic friction between the box and the floor?

$$\mu_k = \frac{F - ma}{mg} = \frac{35.0 - 6.0 \times 0.60}{6.0 \times 9.8} = 0.54$$

Q: A 1280-kg car pulls a 350-kg trailer. The car exerts a horizontal force of $$3.6 \times 10^3$\ \mathrm{N}$$ against the ground. If the coefficient of friction for the trailer is $$0.15$$, what force does the car exert on the trailer?

$$F = m_B a + \mu m_B g = 350 \left(\frac{3.6 \times 10^3$}{1280 + 350}\right) + 0.15 \times 350 \times 9.8 = 1.0 \times 10^3$\ \mathrm{N}$$

Q: A 1200-kg car rounds a turn of radius $$80.0\ \mathrm{m} on a $$flat road. If the coefficient of friction between the tires and the road is $$0.65$$, what is the maximum speed the car can have without skidding?

$$v_{max} = \sqrt{\mu g r} = \sqrt{0.65 \times 9.8 \times 80.0} = 22.6\ \mathrm{m/s}$$

Q: At what minimum speed must a roller coaster be traveling when upside down at the top of a circle of radius $$7.6\ \mathrm{m} so $$that the passengers do not fall out?

$$v_{min} = \sqrt{g r} = \sqrt{9.8 \times 7.6} = 8.6\ \mathrm{m/s}$$

Q: A proposed space station consists of a circular tube with a diameter of $$1.1\ \mathrm{km}. $$What must be the rotation speed (revolutions per day) for the artificial gravity to equal $$1.0\ g$$?

$$f = \frac{1}{2\pi} \sqrt{\frac{g}{r}} \times 86400 = 2.0\ \mathrm{rev/day}$$

Q: A 975-kg sports car crosses the rounded top of a hill (radius $$88.0\ \mathrm{m}) at$$ $$12.0\ \mathrm{m/s}. $$What is the normal force exerted by the road on the car?

$$N = mg - \frac{mv^2$}{r} = 975 \times 9.8 - \frac{975 \times 12.0^2$}{88.0} = 9560\ \mathrm{N}$$

Question 1

A force of 35.0 N is required to start a 6.0-kg box moving across a horizontal concrete floor. What is the coefficient of static friction between the box and the floor?

Accepted Answer

$$\mu_s = \frac{F}{mg} = \frac{35.0}{6.0 	imes 9.8} = 0.60$$

Question 2

A 6.0-kg box is pushed with a constant force of 35.0 N and accelerates at $$0.60\ \mathrm{m/s^2$}. $$What is the coefficient of kinetic friction between the box and the floor?

Accepted Answer

$$\mu_k = \frac{F - ma}{mg} = \frac{35.0 - 6.0 	imes 0.60}{6.0 	imes 9.8} = 0.54$$

Question 3

A 1280-kg car pulls a 350-kg trailer. The car exerts a horizontal force of $$3.6 	imes 10^3$\ \mathrm{N}$$ against the ground. If the coefficient of friction for the trailer is $$0.15$$, what force does the car exert on the trailer?

Accepted Answer

$$F = m_B a + \mu m_B g = 350 \left(\frac{3.6 	imes 10^3$}{1280 + 350}ight) + 0.15 	imes 350 	imes 9.8 = 1.0 	imes 10^3$\ \mathrm{N}$$

Question 4

In the system shown, $$m_B = 2.0\ \mathrm{kg}$$ hangs over the edge, and the coefficient of static friction between $$m_A$$ and the table is $$0.40. $$What is the minimum value of $$m_A$$ that will keep the system from starting to move?

Accepted Answer

$$m_A = \frac{m_B g}{\mu_s g} = \frac{2.0}{0.40} = 5.0\ \mathrm{kg}$$

Question 5

A 1200-kg car rounds a turn of radius $$80.0\ \mathrm{m} on a $$flat road. If the coefficient of friction between the tires and the road is $$0.65$$, what is the maximum speed the car can have without skidding?

Accepted Answer

$$v_{max} = \sqrt{\mu g r} = \sqrt{0.65 	imes 9.8 	imes 80.0} = 22.6\ \mathrm{m/s}$$

Question 6

At what minimum speed must a roller coaster be traveling when upside down at the top of a circle of radius $$7.6\ \mathrm{m} so $$that the passengers do not fall out?

Accepted Answer

$$v_{min} = \sqrt{g r} = \sqrt{9.8 	imes 7.6} = 8.6\ \mathrm{m/s}$$

Question 7

A proposed space station consists of a circular tube with a diameter of $$1.1\ \mathrm{km}. $$What must be the rotation speed (revolutions per day) for the artificial gravity to equal $$1.0\ g$$?

Accepted Answer

$$f = \frac{1}{2\pi} \sqrt{\frac{g}{r}} 	imes 86400 = 2.0\ \mathrm{rev/day}$$

Question 8

A 975-kg sports car crosses the rounded top of a hill (radius $$88.0\ \mathrm{m}) at$$ $$12.0\ \mathrm{m/s}. $$What is the normal force exerted by the road on the car?

Accepted Answer

$$N = mg - \frac{mv^2$}{r} = 975 	imes 9.8 - \frac{975 	imes 12.0^2$}{88.0} = 9560\ \mathrm{N}$$

Question 9

Tarzan (mass $$78\ \mathrm{kg}) $$swings from a vine $$5.2\ \mathrm{m}$$ long. If his arms can exert a force of $$1350\ \mathrm{N}$$, what is the maximum speed he can tolerate at the lowest point of his swing?

Accepted Answer

$$v_{max} = \sqrt{\frac{T_{max} - mg}{m} \ell} = \sqrt{\frac{1350 - 78 	imes 9.8}{78} 	imes 5.2} = 6.2\ \mathrm{m/s}$$

Question 10

In a 'Rotor-ride' at a carnival, people rotate in a vertical cylindrically walled room of radius $$5.5\ \mathrm{m} at$$ $$0.50$$ revolutions per second. What is the minimum coefficient of static friction to keep people from slipping down when the floor drops out?

Accepted Answer

$$\mu_s = \frac{g}{\$$omega^2$$ r}$$, where $$\omega = 2\pi f = 2\pi 	imes 0.50 = 3.14\ \mathrm{rad/s}$$, so $$\mu_s = \frac{9.8}{(3.14)^2$ 	imes 5.5} = 0.18$$

Question 11

A 72-kg water skier is pulled by a boat with a horizontal force of $$240\ \mathrm{N}. $$The coefficient of kinetic friction between the skis and water is $$0.25. $$What is the skier's acceleration?

Accepted Answer

$$a = \frac{F - \mu_k mg}{m} = \frac{240 - 0.25 	imes 72 	imes 9.8}{72} = 0.83\ \mathrm{m/s^2$}$$

Question 12

A sports car crosses the bottom of a valley with a radius of curvature of $$95\ \mathrm{m}. At $$the very bottom, the normal force on the driver is twice his weight. At what speed was the car traveling?

Accepted Answer

$$N = mg + \frac{mv^2$}{r} = 2mg \implies v = \sqrt{g r} = \sqrt{9.8 	imes 95} = 30.5\ \mathrm{m/s}$$

Physics with Calculus: Friction, Circular Motion, and Dynamics Study Guide

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