Q: A 10-kg box is at rest on a horizontal surface. You push it with a constant force of 20 N, and it starts to move. If the coefficient of kinetic friction between the box and the surface is 0.15, what is the acceleration of the box? (Take $$g = 9.8\ \mathrm{m/s^2$}.)$$

$$a = \frac{20 - (0.15 \times 10 \times 9.8)}{10} = 0.53\ \mathrm{m/s^2$}$$

Q: A 5.0-kg box is suspended from a spring. When hung at rest, the spring stretches $$0.050\ \mathrm{m}$$ beyond its relaxed length. What is the spring constant $$k of $$the spring?

$$k = \frac{mg}{\Delta x} = \frac{5.0 \times 9.8}{0.050} = 980\ \mathrm{N/m}$$

Q: A 70-kg astronaut experiences a gravitational force of $$1960\ \mathrm{N} on $$Earth. What is her weight on the Moon, where $$g_{moon} = 1.6\ \mathrm{m/s^2$}$$?

$$F_{moon} = 70 \times 1.6 = 112\ \mathrm{N}$$

Q: A 2.0-kg object is dropped from rest and falls under gravity and air resistance, where the resistive force is $$F_a = -bv$$ with $$b = 4.0\ \mathrm{kg/s}. $$What is the terminal velocity?

$$v_t = \frac{mg}{b} = \frac{2.0 \times 9.8}{4.0} = 4.9\ \mathrm{m/s}$$

Q: A 55-kg skier slides across a rough patch of snow and comes to rest after traveling $$50\ \mathrm{m}. If $$her initial speed is $$27\ \mathrm{km/h}$$, what is the magnitude of the friction force?

$$v_i = 27 \times \frac{1000}{3600} = 7.5\ \mathrm{m/s}$$;\ F = \frac{m v_$$i^2$$}{2d} = \frac{55 \times $$(7.5)^2$$}{2 \times 50} = 31\ \mathrm{N}$

Q: A 0.050-kg egg falls from a height of $$1.0\ \mathrm{m}$$ and lands on a hard floor. What is the impulse delivered to the egg by the floor (ignore air resistance)?

$$v = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 1.0} = 4.43\ \mathrm{m/s}$$;\ J = m v = 0.050 \times 4.43 = 0.22\ \mathrm{kg\cdot m/s}$

Q: A 2.0-kg cart and an 8.0-kg cart are connected by a relaxed spring ($$k = 300\ \mathrm{N/m}). $$You pull the 8.0-kg cart with a constant force so that the spring stretches by $$0.100\ \mathrm{m}. $$What force did you exert?

$$F = kx = 300 \times 0.100 = 30\ \mathrm{N}$$

Question 1

A 10-kg box is at rest on a horizontal surface. You push it with a constant force of 20 N, and it starts to move. If the coefficient of kinetic friction between the box and the surface is 0.15, what is the acceleration of the box? (Take $$g = 9.8\ \mathrm{m/s^2$}.)$$

Accepted Answer

$$a = \frac{20 - (0.15 	imes 10 	imes 9.8)}{10} = 0.53\ \mathrm{m/s^2$}$$

Question 2

A 0.150-kg baseball is pitched at $$40\ \mathrm{m/s}$$ and is hit straight back at $$60\ \mathrm{m/s}. If $$the bat is in contact with the ball for $$2.0\ \mathrm{ms}$$, what is the magnitude of the average force exerted by the bat on the ball?

Accepted Answer

$$F_{avg} = \frac{m(v_{f} - v_{i})}{\Delta t} = \frac{0.150 	imes (60 - (-40))}{0.002} = 7,500\ \mathrm{N}$$

Question 3

A 5.0-kg box is suspended from a spring. When hung at rest, the spring stretches $$0.050\ \mathrm{m}$$ beyond its relaxed length. What is the spring constant $$k of $$the spring?

Accepted Answer

$$k = \frac{mg}{\Delta x} = \frac{5.0 	imes 9.8}{0.050} = 980\ \mathrm{N/m}$$

Question 4

A 2.0-kg cart is pushed by a time-dependent force $$F(t) = 0.5t^2$ (in newtons). What is the speed of the cart at t = 2.0$$\ \mathrm{s}$$ if it starts from rest?

Accepted Answer

$$v = \frac{1}{m} \int_{0}^{2} F(t) dt = \frac{1}{2.0} \int_{0}^{2} 0.5t^2$ dt = \frac{1}{2.0} 	imes 0.5 	imes \frac{8}{3} = 0.67\ \mathrm{m/s}$$

Question 5

A 70-kg astronaut experiences a gravitational force of $$1960\ \mathrm{N} on $$Earth. What is her weight on the Moon, where $$g_{moon} = 1.6\ \mathrm{m/s^2$}$$?

Accepted Answer

$$F_{moon} = 70 	imes 1.6 = 112\ \mathrm{N}$$

Question 6

A 2.0-kg object is dropped from rest and falls under gravity and air resistance, where the resistive force is $$F_a = -bv$$ with $$b = 4.0\ \mathrm{kg/s}. $$What is the terminal velocity?

Accepted Answer

$$v_t = \frac{mg}{b} = \frac{2.0 	imes 9.8}{4.0} = 4.9\ \mathrm{m/s}$$

Question 7

A 30-kg shopping cart is pushed from rest with a constant force of $$12\ \mathrm{N}$$ for $$4.5\ \mathrm{s}. $$How far does it travel in this time?

Accepted Answer

$$a = \frac{12}{30} = 0.4\ \mathrm{m/s^2$}$$;\ x = \frac{1}{2} a $$t^2 = 0.5$$ 	imes 0.4 	imes $$(4.5)^2 = 4.05$$\ \mathrm{m}$

Question 8

A 55-kg skier slides across a rough patch of snow and comes to rest after traveling $$50\ \mathrm{m}. If $$her initial speed is $$27\ \mathrm{km/h}$$, what is the magnitude of the friction force?

Accepted Answer

$$v_i = 27 	imes \frac{1000}{3600} = 7.5\ \mathrm{m/s}$$;\ F = \frac{m v_$$i^2$$}{2d} = \frac{55 	imes $$(7.5)^2$$}{2 	imes 50} = 31\ \mathrm{N}$

Question 9

A 0.050-kg egg falls from a height of $$1.0\ \mathrm{m}$$ and lands on a hard floor. What is the impulse delivered to the egg by the floor (ignore air resistance)?

Accepted Answer

$$v = \sqrt{2gh} = \sqrt{2 	imes 9.8 	imes 1.0} = 4.43\ \mathrm{m/s}$$;\ J = m v = 0.050 	imes 4.43 = 0.22\ \mathrm{kg\cdot m/s}$

Question 10

A 2.0-kg cart and an 8.0-kg cart are connected by a relaxed spring ($$k = 300\ \mathrm{N/m}). $$You pull the 8.0-kg cart with a constant force so that the spring stretches by $$0.100\ \mathrm{m}. $$What force did you exert?

Accepted Answer

$$F = kx = 300 	imes 0.100 = 30\ \mathrm{N}$$

Question 11

A 70-kg student falls toward Earth. What is the acceleration of Earth toward the student? (Earth's inertia is $$5.97 	imes 10$$^{24}$$\ \mathrm{kg}.)$$

Accepted Answer

$$a_{Earth} = \frac{F}{M_{Earth}} = \frac{70 	imes 9.8}{5.97 	imes 10$$^{24}$$} = 1.15 	imes 10$$^{-23}$$\ \mathrm{m/s^2$}$$

Question 12

A 0.27-kg volleyball is hit with a force $$F(t) = a t - b t^2$, where a = 3.0$$ 	imes $$10^5$$\ \mathrm{N/s}$$ and b = 1.0$$ 	imes $$10^8$$\ \mathrm{$$N/s^2$$}$$. If the hand is in contact for 3.0$$\ \mathrm{ms}$$, what is the impulse delivered to the ball?

Accepted Answer

$$J = \int_{0}^{0.003} (3.0 	imes 10^5$ t - 1.0 	imes 10^8$ t^2$) dt = [1.5 	imes 10^5$ t^2$ - (1/3) 	imes 1.0 	imes 10^8$ t^3$]_{0}^{0.003} = 1.35\ \mathrm{N\cdot s}$$

Question 13

A 2.3-kg ball is dropped from $$2.0\ \mathrm{m}$$ and bounces back to $$1.7\ \mathrm{m}. $$What is the impulse delivered to the ball by the plate?

Accepted Answer

$$v_{down} = \sqrt{2 	imes 9.8 	imes 2.0} = 6.26\ \mathrm{m/s}$$;\ v_{up} = \sqrt{2 	imes 9.8 	imes 1.7} = 5.78\ \mathrm{m/s}$$;\ J = m(v_{up} + v_{down}) = 2.3 	imes (6.26 + 5.78) = 27.8\ \mathrm{kg\cdot m/s}$$

Question 14

A 0.70-kg pillow is pushed across a bed with a constant force of $$10\ \mathrm{N}$$, while the bed provides a frictional force of $$6.0\ \mathrm{N}. $$What is the acceleration of the center of mass of the pillow?

Accepted Answer

$$a = \frac{10 - 6}{0.70} = 5.7\ \mathrm{m/s^2$}$$

Question 15

A 50-kg pneumatic hammer moves vertically with $$x(t) = at^2$ - bt^3$, where a = 15$$\ \mathrm{$$m/s^2$$}$$ and b = 20$$\ \mathrm{$$m/s^3$$}$$. What is the expression for the x$-component of the vector sum of forces on the hammer as a function of time?

Accepted Answer

$$F_x = m \frac{d^2$x}{dt^2$} = 50 	imes (2a - 6bt) = 100a - 300bt$$

Question 16

A 5.0-kg block is suspended from a spring scale and slowly lowered onto a vertical spring. If the scale reads $$40\ \mathrm{N}$$ when the bottom spring is compressed $$30\ \mathrm{mm}$$, what is the spring constant $$k of $$the bottom spring?

Accepted Answer

$$k = \frac{mg - F_{scale}}{\Delta x} = \frac{49 - 40}{0.030} = 300\ \mathrm{N/m}$$

Question 17

A 1100-kg car slows from $$85\ \mathrm{km/h} to $$rest in $$20\ \mathrm{s}. $$What is the average force exerted on the car?

Accepted Answer

$$v_i = 85 	imes \frac{1000}{3600} = 23.6\ \mathrm{m/s}$$;\ F_{avg} = \frac{m(v_f - v_i)}{\Delta t} = \frac{1100 	imes (0 - 23.6)}{20} = -1295\ \mathrm{N}$

Question 18

A 4.0-kg block and a 2.0-kg block are connected by a spring ($$k = 300\ \mathrm{N/m}). $$The 4.0-kg block is pushed with $$50\ \mathrm{N}$$ and the 2.0-kg block with $$20\ \mathrm{N}$$ for $$0.50\ \mathrm{s}. $$What is the acceleration of the center of mass during this time?

Accepted Answer

$$a_{cm} = \frac{F_{total}}{m_{total}} = \frac{50 + 20}{4.0 + 2.0} = 11.7\ \mathrm{m/s^2$}$$

Chapter 8: Force – Step-by-Step Physics Study Guidance

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