Q: A 2 kg block is attached to a horizontal spring with a spring constant $$k = 200 \ \mathrm{N/m}. $$The block is pulled 0.1 m from equilibrium and released from rest. What is the maximum speed of the block during its motion?

$$ $$\sqrt{\frac{k}{m}}A = \sqrt{\frac{200}{2}} \times 0.1 = \sqrt{100} \times 0.1 = 10 \times 0.1 = 1.0 \ \mathrm{m/s}$

Q: A 0.5 kg ball is swung in a horizontal circle of radius 1.5 m at a constant speed of 3 m/s. What is the magnitude of the net force acting on the ball?

$$F = m a_c = m \frac{v^2$}{r} = 0.5 \times \frac{9}{1.5} = 3 \ \mathrm{N}$$

Q: A 4 kg object is dropped from a height of 10 m. Ignoring air resistance, what is its speed just before hitting the ground?

$$v = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 10} = \sqrt{196} = 14 \ \mathrm{m/s}$$

Q: A 0.2 kg mass attached to a spring oscillates with a period of 0.9 s. What is the spring constant $$k$$?

$$k = \frac{4\pi^2$ m}{T^2$} = \frac{4\pi^2$ \times 0.2}{0.81} \approx 9.75 \ \mathrm{N/m}$$

Q: A 5 kg box is pulled up a 30° incline (frictionless) at constant speed for 4 m. What is the work done by the pulling force?

$$W = mg \sin(30^\circ) \times 4 = 5 \times 9.8 \times 0.5 \times 4 = 98 \ \mathrm{J}$$

Q: A uniform beam of length 6 m and mass 20 kg is supported at its ends. A 60 kg person stands 2 m from the left end. What is the force on the left support?

$$F_{left} = 20 \times 9.8 \times 0.5 + 60 \times 9.8 \times \frac{4}{6} = 98 + 392 \times 0.666 = 98 + 261 = 359 \ \mathrm{N}$$

Q: A car of mass 1200 kg rounds a curve of radius 50 m at a speed of 15 m/s. What is the minimum coefficient of static friction required to prevent skidding?

$$\mu_s = \frac{v^2$}{rg} = \frac{225}{50 \times 9.8} = \frac{225}{490} \approx 0.46$$

Question 1

A 2 kg block is attached to a horizontal spring with a spring constant $$k = 200 \ \mathrm{N/m}. $$The block is pulled 0.1 m from equilibrium and released from rest. What is the maximum speed of the block during its motion?

Accepted Answer

$$
$$\sqrt{\frac{k}{m}}A = \sqrt{\frac{200}{2}} 	imes 0.1 = \sqrt{100} 	imes 0.1 = 10 	imes 0.1 = 1.0 \ \mathrm{m/s}$

Question 2

A disk of mass $$M = 3 \ \mathrm{kg}$$ and radius $$R = 0.2 \ \mathrm{m}$$ rotates about its center with angular velocity $$\omega = 10 \ \mathrm{rad/s}. $$What is its rotational kinetic energy?

Accepted Answer

$$K_{rot} = \frac{1}{2} I \$$omega^2$$ = \frac{1}{2} \left(\frac{1}{2}MR^2$ight) \$$omega^2$$ = \frac{1}{4}MR^2$\$$omega^2$$ = \frac{1}{4} 	imes 3 	imes (0.2)^2$ 	imes 100 = 3 \ \mathrm{J}$$

Question 3

A 0.5 kg ball is swung in a horizontal circle of radius 1.5 m at a constant speed of 3 m/s. What is the magnitude of the net force acting on the ball?

Accepted Answer

$$F = m a_c = m \frac{v^2$}{r} = 0.5 	imes \frac{9}{1.5} = 3 \ \mathrm{N}$$

Question 4

A 4 kg object is dropped from a height of 10 m. Ignoring air resistance, what is its speed just before hitting the ground?

Accepted Answer

$$v = \sqrt{2gh} = \sqrt{2 	imes 9.8 	imes 10} = \sqrt{196} = 14 \ \mathrm{m/s}$$

Question 5

A 0.2 kg mass attached to a spring oscillates with a period of 0.9 s. What is the spring constant $$k$$?

Accepted Answer

$$k = \frac{4\pi^2$ m}{T^2$} = \frac{4\pi^2$ 	imes 0.2}{0.81} \approx 9.75 \ \mathrm{N/m}$$

Question 6

A 5 kg box is pulled up a 30° incline (frictionless) at constant speed for 4 m. What is the work done by the pulling force?

Accepted Answer

$$W = mg \sin(30^\circ) 	imes 4 = 5 	imes 9.8 	imes 0.5 	imes 4 = 98 \ \mathrm{J}$$

Question 7

A uniform beam of length 6 m and mass 20 kg is supported at its ends. A 60 kg person stands 2 m from the left end. What is the force on the left support?

Accepted Answer

$$F_{left} = 20 	imes 9.8 	imes 0.5 + 60 	imes 9.8 	imes \frac{4}{6} = 98 + 392 	imes 0.666 = 98 + 261 = 359 \ \mathrm{N}$$

Question 8

A 0.1 kg bullet moving at 400 m/s embeds itself in a 2 kg block at rest on a frictionless surface. What is the final velocity of the block-bullet system?

Accepted Answer

$$v_f = \frac{m_{bullet} v_{bullet}}{m_{bullet} + m_{block}} = \frac{0.1 	imes 400}{2.1} \approx 19.05 \ \mathrm{m/s}$$

Question 9

A 1.5 m long simple pendulum swings with a maximum angular displacement of 10°. What is its period?

Accepted Answer

$$T = 2\pi \sqrt{\frac{L}{g}} = 2\pi \sqrt{\frac{1.5}{9.8}} \approx 2.46 \ \mathrm{s}$$

Question 10

A car of mass 1200 kg rounds a curve of radius 50 m at a speed of 15 m/s. What is the minimum coefficient of static friction required to prevent skidding?

Accepted Answer

$$\mu_s = \frac{v^2$}{rg} = \frac{225}{50 	imes 9.8} = \frac{225}{490} \approx 0.46$$

Question 11

A 0.5 kg mass is attached to a string and swung in a vertical circle of radius 2 m. What is the minimum speed at the top of the circle to keep the string taut?

Accepted Answer

$$v_{min} = \sqrt{g r} = \sqrt{9.8 	imes 2} = \sqrt{19.6} \approx 4.43 \ \mathrm{m/s}$$

Question 12

A 10 kg object is moving at 5 m/s. What is its momentum?

Accepted Answer

$$p = m v = 10 	imes 5 = 50 \ \mathrm{kg \cdot m/s}$$

Question 13

A force $$F = 10 \ \mathrm{N} is $$applied tangentially to a wheel of radius $$0.5 \ \mathrm{m}$$ for $$2 \ \mathrm{s}. $$What is the angular impulse delivered to the wheel?

Accepted Answer

$$J = 	au \Delta t = F r \Delta t = 10 	imes 0.5 	imes 2 = 10 \ \mathrm{N \cdot m \cdot s}$$

Question 14

A 2 kg mass oscillates on a spring with amplitude 0.3 m and $$k = 50 \ \mathrm{N/m}. $$What is the total mechanical energy of the system?

Accepted Answer

$$E = \frac{1}{2} k A^2$ = 0.5 	imes 50 	imes (0.3)^2$ = 2.25 \ \mathrm{J}$$

Question 15

A 3 kg object moves in a circle of radius 4 m with angular velocity $$2 \ \mathrm{rad/s}. $$What is its angular momentum about the center?

Accepted Answer

$$L = m v r = m (r \omega) r = m r^2$ \omega = 3 	imes 16 	imes 2 = 96 \ \mathrm{kg \cdot m^2$/s}$$

Question 16

A 0.5 kg mass is attached to a spring and oscillates with $$x(t) = 0.2 \cos(5t). $$What is the maximum acceleration of the mass?

Accepted Answer

$$a_{max} = \$$omega^2$$ A = 25 	imes 0.2 = 5 \ \mathrm{m/s^2$}$$

Question 17

A 2 kg mass is attached to a string and swung in a horizontal circle of radius 1 m at 4 m/s. What is the tension in the string?

Accepted Answer

$$T = m \frac{v^2$}{r} = 2 	imes \frac{16}{1} = 32 \ \mathrm{N}$$

Question 18

A 0.3 kg mass is attached to a spring with $$k = 12 \ \mathrm{N/m}. $$What is the frequency of oscillation?

Accepted Answer

$$f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} = \frac{1}{2\pi} \sqrt{\frac{12}{0.3}} = \frac{1}{2\pi} \sqrt{40} \approx 1.01 \ \mathrm{Hz}$$

PHYS 201 Equation Sheet: Chapters 1–11 (Mechanics, Rotational Motion, and Periodic Motion)

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