Q: The scalar product of vector $$\vec{A} = 3.00\hat{i} + 2.00\hat{j}$$ and vector $$\vec{B} = 1.00\hat{i} + 4.00\hat{j} is$$:

$$3.00 \times 1.00 + 2.00 \times 4.00 = 11.00$$

Q: The acceleration of an object as a function of time is given by $$a(t) = (10.0\ \mathrm{m/s^2$})t$$, where $$t is in $$seconds. What is the velocity of the object at time $$t = 4.00\ \mathrm{s} if it $$starts at rest?

$$v = \$$int_0$$^4 10t\ dt = 5t^2$ \Big|_0^4$ = 5 \times 16 = 80\ \mathrm{m/s}$$

Q: A ball is thrown horizontally from a height of $$2.00\ \mathrm{m}$$ with a speed of $$5.00\ \mathrm{m/s}. $$How long does it take to hit the ground?

$$t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2 \times 2.00}{9.8}} \approx 0.64\ \mathrm{s}$$

Q: A $$20\ \mathrm{kg}$$ box is pushed across a rough horizontal surface with a constant force $$F = 100\ \mathrm{N}$$ for $$8.00\ \mathrm{m}. $$The box's speed changes from $$0.500\ \mathrm{m/s} to$$ $$2.00\ \mathrm{m/s}. $$What is the work done by friction?

$$W_{fric} = Fd - \Delta KE = 100 \times 8.00 - \frac{1}{2} \times 20 \times (2.00^2$ - 0.500^2$) = 800 - 37.5 = 762.5\ \mathrm{J}$$

Q: A $$0.200\ \mathrm{kg}$$ ball collides with a wall, changing velocity from $$0.250\ \mathrm{m/s} to$$ $$-0.250\ \mathrm{m/s} in$$ $$0.0200\ \mathrm{s}. $$What is the magnitude of the average force exerted by the wall?

$$F_{avg} = \frac{\Delta p}{\Delta t} = \frac{0.200 \times (0.250 - (-0.250))}{0.0200} = 5.00\ \mathrm{N}$$

Q: A solid disk of radius $$1.00\ \mathrm{m}$$ and mass $$2.50\ \mathrm{kg}$$ rolls without slipping down a $$2.00\ \mathrm{m}$$ high incline. What is its speed at the bottom?

$$v = \sqrt{\frac{4gh}{3}} = \sqrt{\frac{4 \times 9.8 \times 2.00}{3}} \approx 5.11\ \mathrm{m/s}$$

Q: A $$0.200\ \mathrm{kg}$$ mass is attached to a string and swung in a horizontal circle of radius $$0.50\ \mathrm{m} at$$ $$4.0\ \mathrm{m/s}. $$What is the tension in the string?

$$T = m \frac{v^2$}{r} = 0.200 \times \frac{4.0^2$}{0.50} = 6.4\ \mathrm{N}$$

Question 1

A vector $$\vec{Z} is $$given by $$\vec{Z} = 3\hat{i} + 4\hat{j}. $$What is the magnitude of $$\vec{Z}$$?

Accepted Answer

$$\sqrt{3^2$ + 4^2$} = 5$$

Question 2

The scalar product of vector $$\vec{A} = 3.00\hat{i} + 2.00\hat{j}$$ and vector $$\vec{B} = 1.00\hat{i} + 4.00\hat{j} is$$:

Accepted Answer

$$3.00 	imes 1.00 + 2.00 	imes 4.00 = 11.00$$

Question 3

An object starts at rest and moves with a constant velocity of $$2.0\ \mathrm{m/s}$$ for $$4.0\ \mathrm{s}$$, then slows to rest in $$2.0\ \mathrm{s}. $$Which graph best represents the velocity vs. time?

Accepted Answer

A graph with a horizontal line at $$2.0\ \mathrm{m/s}$$ for $$4.0\ \mathrm{s}$$, then a straight line down to zero at $$6.0\ \mathrm{s}.$$

Question 4

The acceleration of an object as a function of time is given by $$a(t) = (10.0\ \mathrm{m/s^2$})t$$, where $$t is in $$seconds. What is the velocity of the object at time $$t = 4.00\ \mathrm{s} if it $$starts at rest?

Accepted Answer

$$v = \$$int_0$$^4 10t\ dt = 5t^2$ \Big|_0^4$ = 5 	imes 16 = 80\ \mathrm{m/s}$$

Question 5

A ball is thrown horizontally from a height of $$2.00\ \mathrm{m}$$ with a speed of $$5.00\ \mathrm{m/s}. $$How long does it take to hit the ground?

Accepted Answer

$$t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2 	imes 2.00}{9.8}} \approx 0.64\ \mathrm{s}$$

Question 6

A $$20\ \mathrm{kg}$$ box is pushed across a rough horizontal surface with a constant force $$F = 100\ \mathrm{N}$$ for $$8.00\ \mathrm{m}. $$The box's speed changes from $$0.500\ \mathrm{m/s} to$$ $$2.00\ \mathrm{m/s}. $$What is the work done by friction?

Accepted Answer

$$W_{fric} = Fd - \Delta KE = 100 	imes 8.00 - \frac{1}{2} 	imes 20 	imes (2.00^2$ - 0.500^2$) = 800 - 37.5 = 762.5\ \mathrm{J}$$

Question 7

A $$0.200\ \mathrm{kg}$$ ball collides with a wall, changing velocity from $$0.250\ \mathrm{m/s} to$$ $$-0.250\ \mathrm{m/s} in$$ $$0.0200\ \mathrm{s}. $$What is the magnitude of the average force exerted by the wall?

Accepted Answer

$$F_{avg} = \frac{\Delta p}{\Delta t} = \frac{0.200 	imes (0.250 - (-0.250))}{0.0200} = 5.00\ \mathrm{N}$$

Question 8

A grinding wheel rotates at $$25\ \mathrm{rad/s}$$ and slows to rest in $$10.0\ \mathrm{s}. $$What is the angular acceleration?

Accepted Answer

$$\alpha = \frac{\omega_f - \omega_i}{t} = \frac{0 - 25}{10.0} = -2.5\ \mathrm{$$rad/s^2$$}$$

Question 9

A solid disk of radius $$1.00\ \mathrm{m}$$ and mass $$2.50\ \mathrm{kg}$$ rolls without slipping down a $$2.00\ \mathrm{m}$$ high incline. What is its speed at the bottom?

Accepted Answer

$$v = \sqrt{\frac{4gh}{3}} = \sqrt{\frac{4 	imes 9.8 	imes 2.00}{3}} \approx 5.11\ \mathrm{m/s}$$

Question 10

A potter's wheel, with rotational inertia $$4.0\ \mathrm{kg\cdot m^2$}$$, is spinning freely at $$6.0\ \mathrm{rad/s}. $$The potter drops a lump of clay of mass $$2.0\ \mathrm{kg} at a $$radius of $$0.50\ \mathrm{m}. $$What is the final angular velocity?

Accepted Answer

$$\omega_f = \frac{I_i \omega_i}{I_i + m_{clay} r^2$} = \frac{4.0 	imes 6.0}{4.0 + 2.0 	imes 0.5^2$} = \frac{24.0}{4.5} \approx 5.33\ \mathrm{rad/s}$$

Question 11

A $$0.200\ \mathrm{kg}$$ mass is attached to a string and swung in a horizontal circle of radius $$0.50\ \mathrm{m} at$$ $$4.0\ \mathrm{m/s}. $$What is the tension in the string?

Accepted Answer

$$T = m \frac{v^2$}{r} = 0.200 	imes \frac{4.0^2$}{0.50} = 6.4\ \mathrm{N}$$

Question 12

A $$5.0\ \mathrm{kg}$$ box is pushed down a ramp inclined at $$30^\circ$$ with a force of $$20\ \mathrm{N}. $$The coefficient of kinetic friction is $$0.20. $$What is the acceleration of the box?

Accepted Answer

$$a = \frac{F_{net}}{m} = \frac{20 + 5.0 	imes 9.8 	imes \sin 30^\circ - 0.20 	imes 5.0 	imes 9.8 	imes \cos 30^\circ}{5.0} \approx 3.1\ \mathrm{m/s^2$}$$

Question 13

A $$2.0\ \mathrm{kg}$$ block is attached to a spring with $$k = 50\ \mathrm{N/m}$$ and oscillates with amplitude $$0.10\ \mathrm{m}. $$What is the maximum speed?

Accepted Answer

$$v_{max} = A \sqrt{\frac{k}{m}} = 0.10 	imes \sqrt{\frac{50}{2.0}} = 0.10 	imes 5.0 = 0.50\ \mathrm{m/s}$$

Question 14

A $$100\ \mathrm{N}$$ traffic light is supported by two ropes at $$37^\circ$$ from the horizontal. What is the tension in each rope?

Accepted Answer

$$T = \frac{W}{2 \sin 37^\circ} = \frac{100}{2 	imes 0.6018} \approx 83\ \mathrm{N}$$

Question 15

A $$0.200\ \mathrm{kg}$$ mass and a $$0.100\ \mathrm{kg}$$ mass are connected by a light string over a frictionless pulley. If the system is released from rest, what is the acceleration of the masses?

Accepted Answer

$$a = \frac{(m_1$ - m_2$)g}{m_1$ + m_2$} = \frac{(0.200 - 0.100) 	imes 9.8}{0.200 + 0.100} = \frac{0.100 	imes 9.8}{0.300} \approx 3.27\ \mathrm{m/s^2$}$$

Question 16

A $$0.200\ \mathrm{kg}$$ mass is dropped from a height of $$2.0\ \mathrm{m}. $$What is its speed just before hitting the ground?

Accepted Answer

$$v = \sqrt{2gh} = \sqrt{2 	imes 9.8 	imes 2.0} \approx 6.26\ \mathrm{m/s}$$

Question 17

A $$0.500\ \mathrm{kg}$$ marble slides to the left at $$0.400\ \mathrm{m/s}$$ and collides elastically with a $$0.200\ \mathrm{kg}$$ marble at rest. What is the velocity of the $$0.200\ \mathrm{kg}$$ marble after the collision?

Accepted Answer

$$v_2$$ = \frac{2m_1}{$$m_1$$ + $$m_2$$} $$v_1$$ = \frac{2 	imes 0.500}{0.500 + 0.200} 	imes 0.400 \approx 0.571\ \mathrm{m/s}$$

PHYS 220A – Fall 2023 Final Exam Study Notes

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