Q: A vector $$\vec{Z} is $$given by $$\vec{Z} = \vec{A} + \vec{B}. If $$the magnitude of vector $$\vec{A} is 10 $$units and the magnitude of vector $$\vec{B} is 7.0 $$units, and the angle between them is $$60^\circ$$, what is the magnitude of vector $$\vec{Z}$$?

$$\sqrt{10^2$ + 7^2$ + 2 \times 10 \times 7 \times \cos 60^\circ} = 15.2$$

Q: The scalar product of vector $$\vec{A} = 2.00\hat{i} + 2.00\hat{j}$$ and vector $$\vec{B} = 10\hat{i} is$$:

$$2.00 \times 10 + 2.00 \times 0 = 20.0$$

Q: The acceleration of an object as a function of time is given by $$a(t) = (10.0\,\text{m/s}^2)t$$, where $$t is in $$seconds. If the object is at rest at time $$t = 0.0 s$$, what is the velocity of the object at time $$t = 4.00 s$$?

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

Q: A ball is inside the end of a tube of negligible mass. The ball is spun in a circle with a radius 2.00 m making 7.00 revolutions every 10.0 seconds. What is the magnitude of the acceleration of the ball?

$$a = \$$omega^2$$ r = \left(\frac{2\pi \times 7}{10}\$$right)^2$$ \times 2.00 = 27.2\,\text{m/s}^2$$

Q: A 7.8-kg object is acted on by two forces. One of the forces is 18.0 N acting toward the east. Which of the following forces results in the net force of the acceleration of the object is 2.0 m/s$$^2$$ toward the east?

$$F = ma = 7.8 \times 2.0 = 15.6\,\text{N}$$, so the other force is $$18.0 - 15.6 = 2.4\,\text{N}$$ east.

Q: An object of mass 2.0 kg moves along the x-axis. The potential energy curve as a function of position along the x-axis is shown in the figure. If the kinetic energy of the object at the origin is 12 J, what will be its kinetic energy at x = 2.0 m if the potential energy at x = 2.0 m is 8 J?

$$K_f = K_i + U_i - U_f = 12 + 0 - 8 = 4\,\text{J}$$

Question 1

A vector $$\vec{Z} is $$given by $$\vec{Z} = \vec{A} + \vec{B}. If $$the magnitude of vector $$\vec{A} is 10 $$units and the magnitude of vector $$\vec{B} is 7.0 $$units, and the angle between them is $$60^\circ$$, what is the magnitude of vector $$\vec{Z}$$?

Accepted Answer

$$\sqrt{10^2$ + 7^2$ + 2 	imes 10 	imes 7 	imes \cos 60^\circ} = 15.2$$

Question 2

The scalar product of vector $$\vec{A} = 2.00\hat{i} + 2.00\hat{j}$$ and vector $$\vec{B} = 10\hat{i} is$$:

Accepted Answer

$$2.00 	imes 10 + 2.00 	imes 0 = 20.0$$

Question 3

A student starts moving with a constant velocity of 2 m/s toward the east. After 3.0 s, the student stops for 1.0 s, then continues with a velocity of 1 m/s toward the west for the next 2.0 s. Which graph below best represents the position vs. time for this student?

Accepted Answer

Graph (I): Position increases linearly, plateaus, then decreases linearly.

Question 4

The acceleration of an object as a function of time is given by $$a(t) = (10.0\,	ext{m/s}^2)t$$, where $$t is in $$seconds. If the object is at rest at time $$t = 0.0 s$$, what is the velocity of the object at time $$t = 4.00 s$$?

Accepted Answer

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

Question 5

While an object is in projectile motion (with upward being positive) with no air resistance:

Accepted Answer

The horizontal component of its velocity remains constant and the vertical component of its velocity changes due to gravity.

Question 6

A ball is inside the end of a tube of negligible mass. The ball is spun in a circle with a radius 2.00 m making 7.00 revolutions every 10.0 seconds. What is the magnitude of the acceleration of the ball?

Accepted Answer

$$a = \$$omega^2$$ r = \left(\frac{2\pi 	imes 7}{10}\$$right)^2$$ 	imes 2.00 = 27.2\,	ext{m/s}^2$$

Question 7

A 7.8-kg object is acted on by two forces. One of the forces is 18.0 N acting toward the east. Which of the following forces results in the net force of the acceleration of the object is 2.0 m/s$$^2$$ toward the east?

Accepted Answer

$$F = ma = 7.8 	imes 2.0 = 15.6\,	ext{N}$$, so the other force is $$18.0 - 15.6 = 2.4\,	ext{N}$$ east.

Question 8

You push downward on a box at an angle $$25^\circ$$ below the horizontal with a force of 750 N. If the box is on a flat horizontal surface for which the coefficient of static friction with the box is 0.36, what is the mass of the heaviest box you will be able to move?

Accepted Answer

$$f_s = \mu_s N$$, $$N = mg + 750 \sin 25^\circ$$, $$f_s = 0.36 (mg + 750 \sin 25^\circ) = 750 \cos 25^\circ$$, $$m = 134\,	ext{kg}$$

Question 9

A 0.20-kg hockey puck is launched up a ramp that is inclined at a $$30^\circ$$ angle. The coefficient of kinetic friction between the hockey puck and the ramp is $$\mu_k = 0.30. $$The puck's initial speed is 4.0 m/s. How long does it take for the hockey puck to come to a momentary stop (before it slides back down again)?

Accepted Answer

$$a = -g \sin 30^\circ - \mu_k g \cos 30^\circ$$, $$v = v_0$ + at$$, $$0 = 4.0 + at$$, $$t = -\frac{4.0}{a} = 0.77\,	ext{s}$$

Question 10

An object of mass 2.0 kg moves along the x-axis. The potential energy curve as a function of position along the x-axis is shown in the figure. If the kinetic energy of the object at the origin is 12 J, what will be its kinetic energy at x = 2.0 m if the potential energy at x = 2.0 m is 8 J?

Accepted Answer

$$K_f = K_i + U_i - U_f = 12 + 0 - 8 = 4\,	ext{J}$$

Question 11

In the figure, a constant external force $$F = 104 N is $$applied to a 20.0 kg box, which is on a rough horizontal surface. While the force pushes the box a distance of 8.00 m, the speed changes from 0.500 m/s to 2.00 m/s. The work done by friction during this process is closest to:

Accepted Answer

$$W_{fric} = Fd - \Delta KE = 104 	imes 8 - \frac{1}{2} 	imes 20 (2^2$ - 0.5^2$) = 832 - 37.5 = 794.5\,	ext{J}$$

Question 12

Two objects of the same mass move along the same line in opposite directions. The first mass is moving at 4.0 m/s, the second at 3.0 m/s. After collision, they stick together and move at 1.0 m/s in the direction of the first mass before the collision. What was the speed of the second mass before the collision?

Accepted Answer

$$m_1$$ $$v_1$$ + $$m_2$$ $$v_2$$ = ($$m_1$$ + $$m_2) v_f$, v_2$ = \frac{(m_1$ + m_2$) v_f - m_1$ v_1$}{m_2$} = \frac{2m 	imes 1 - m 	imes 4}{m} = -2\,	ext{m/s}$$

Question 13

During a collision with a wall, the velocity of a 0.250-kg ball changes from 20.0 m/s toward the wall to 16.0 m/s away from the wall. If the contact with the wall was 0.010 s, what was the magnitude of the average force applied to the ball?

Accepted Answer

$$F = \frac{\Delta p}{\Delta t} = \frac{0.25 	imes (16 - (-20))}{0.01} = 900\,	ext{N}$$

Question 14

A machinist uses a grinding wheel that is rotating at an angular velocity of 24 rad/s. Upon completing her job, the machinist turns the power off. The wheel decelerates uniformly at 1.5 rad/s$$^2$$ until the wheel stops. What is the time interval of angular deceleration (slowing down) to rest?

Accepted Answer

$$t = \frac{\omega}{\alpha} = \frac{24}{1.5} = 16\,	ext{s}$$

Question 15

A uniform solid disk of radius 1.00 m and mass 2.50 kg rolls without slipping to the bottom of an inclined plane. The angular velocity of the disk is 3.35 rad/s at the bottom of the plane. What is the height of the plane? The moment of inertia of a solid disk is $$I = \frac{1}{2} m r^2$.

Accepted Answer

$$mgh = \frac{1}{2} m v^2$ + \frac{1}{2} I \$$omega^2$$, v = r$$ \omega$$, h$$ = \frac{1}{2} $$r^2$$ \$$omega^2 / g = 0.57$$\,	ext{m}$$

Question 16

A potter's wheel, with rotational inertia 40.0 kg·m$$^2$$, is spinning freely at 40 rpm. The potter drops a lump (4.0 kg) of clay, which sticks at a distance 0.20 m from the rotational axis. What is the subsequent angular speed of the wheel and clay in rpm after the mass of the clay?

Accepted Answer

$$I_{total} = 40 + 4 	imes 0.2^2$ = 40.16$$, $$\omega_f = \omega_i \frac{I_i}{I_f} = 40 	imes \frac{40}{40.16} = 39.8\,	ext{rpm}$$

Question 17

A light triangular plate (1.0 kg) is in a horizontal plane. Three forces, $$F_1 = 6.0$ N, F_2$ = 9.0 N$$, and $$F_3 = 7.0$ N, act on the plate, which is pivoted about a vertical axis through point O. The sum of the torques about the vertical axis through point O, acting on the plate due to the forces, is closest to:

Accepted Answer

$$	au = F_1$ r_1$ + F_2$ r_2$ - F_3$ r_3$ = 4.4\,	ext{Nm}$$

Question 18

A traffic light weighing 100 N is supported by two ropes as shown in the figure. The tensions in the ropes are closest to:

Accepted Answer

$$T = \frac{100}{2 \cos 37^\circ} = 79.6\,	ext{N}$$

PHYS 2204 – Final Exam Study Notes: Mechanics, Energy, and Rotational Motion

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