Question 1

Five forces act on a rod that is free to pivot at point P, as shown in the figure. Which of these forces is producing a counter-clockwise torque about point P? (There could be more than one correct choice.) 
A) force A
B) force B
C) force C
D) force D
E) force E

Accepted Answer

C

Question 2

Three masses are located in the x-y plane as follows: a mass of 6 kg is at (0 m, 0 m), a mass of 4 kg is at (3 m, 0 m), and a mass of 2 kg is at (0 m, 3 m). Where is the center of mass (or center of gravity) of the system?
A) (1 m, 2 m)
B) (2 m, 1 m)
C) (1 m, 1 m)
D) (1 m, 0.5 m)
E) (0.5 m, 1 m)

Accepted Answer

First, we can find the x-coordinate of the center of mass by taking the weighted average of the x-coordinates of the three masses. That is,

x_cm = (6 kg)(0 m) + (4 kg)(3 m) + (2 kg)(0 m) / (6 kg + 4 kg + 2 kg) 
= 12 m / 12 kg 
= 1 m

Similarly, we can find the y-coordinate of the center of mass by taking the weighted average of the y-coordinates of the three masses. That is,

y_cm = (6 kg)(0 m) + (4 kg)(0 m) + (2 kg)(3 m) / (6 kg + 4 kg + 2 kg) 
= 6 m / 12 kg 
= 0.5 m

Therefore, the center of mass is located at (1 m, 0.5 m), which corresponds to choice D.

Question 3

In the figure, four point masses are placed as shown. Assume that all the numbers in the figure are accurate to two significant figures. What are the x and y coordinates of the center of mass (or center of gravity) of this arrangement? 
A) (2.2 m, 2.6 m)
B) (2.2 m, 2.7 m)
C) (2.3 m, 2.6 m)
D) (2.3 m, 2.7 m)
E) (2.3 m, 2.8 m)

Accepted Answer

E

Question 4

A 2.0-m rope is lying on a table. You pick up one end and start raising it vertically. How high above the table is the center of mass (or center of gravity) of the rope when half of the rope has lifted off the table?
A) 0.25 m
B) 0.50 m
C) 0.75 m
D) 1.0 m
E) 1.5 m

Accepted Answer

When half of the rope is lifted off the table, the center of mass of the lifted portion is at the midpoint of that half, which is 0.5 meters from the end being lifted. Since this is half of the rope, the midpoint is 1 meter from the end, making the center of mass 0.25 meters above the table.

Question 5

Three small masses are positioned as follows: 2.0 kg at (0.0 m, 0.0 m), 2.0 kg at (2.0 m, 0.0 m), and 4.0 kg at (2.0 m, 1.0 m). Determine the coordinates of the center of mass (or center of gravity) of this system.
A) (0.50 m, 1.5 m)
B) (1.5 m, 0.50 m)
C) (2.5 m, 1.5 m)
D) (2.5 m, 0.50 m)

Accepted Answer

The center of mass $x_{cm}$ and $y_{cm}$ can be calculated using the formulas:$$x_{cm} = \frac{\sum m_i x_i}{\sum m_i}$$$$y_{cm} = \frac{\sum m_i y_i}{\sum m_i}$$Given the masses and their coordinates:- Mass 1: $2.0 \, \text{kg}$ at $(0.0 \, \text{m}, 0.0 \, \text{m})$- Mass 2: $2.0 \, \text{kg}$ at $(2.0 \, \text{m}, 0.0 \, \text{m})$- Mass 3: $4.0 \, \text{kg}$ at $(2.0 \, \text{m}, 1.0 \, \text{m})$Calculating $x_{cm}$:$$x_{cm} = \frac{(2.0 \times 0.0) + (2.0 \times 2.0) + (4.0 \times 2.0)}{2.0 + 2.0 + 4.0} = \frac{0 + 4 + 8}{8} = \frac{12}{8} = 1.5 \, \text{m}$$Calculating $y_{cm}$:$$y_{cm} = \frac{(2.0 \times 0.0) + (2.0 \times 0.0) + (4.0 \times 1.0)}{2.0 + 2.0 + 4.0} = \frac{0 + 0 + 4}{8} = \frac{4}{8} = 0.5 \, \text{m}$$Therefore, the coordinates of the center of mass are $(1.5 \, \text{m}, 0.5 \, \text{m})$.

Question 6

When a rigid object rotates about a fixed axis, what is true about all the points in the object? (There could be more than one correct choice.)
A) They all have the same angular speed.
B) They all have the same tangential speed.
C) They all have the same angular acceleration.
D) They all have the same tangential acceleration.
E) They all have the same radial acceleration.

Accepted Answer

The answer of When a rigid object rotates about a...

Question 7

The figure shows scale drawings of four objects, each of the same mass and uniform thickness, with the mass distributed uniformly. Which one has the greatest moment of inertia when rotated about an axis perpendicular to the plane of the drawing at point P? 
A) A
B) B
C) C
D) D
E) The moment of inertia is the same for all of these objects.

Accepted Answer

The moment of inertia of an object depends on its shape and mass distribution. Object B has the majority of its mass distributed farthest from point P, while the other three objects have their mass distributed in a more uniform manner. This means that Object B will have the greatest moment of inertia when rotated about an axis perpendicular to the plane of the drawing at point P.

Question 8

As shown in the figure, a given force is applied to a rod in several different ways. In which case is the torque about the pivot P due to this force the greatest? 
1.
2.
3.
4.
5.
A) 1
B) 2
C) 3
D) 4
E) 5

Accepted Answer

The answer of As shown in the figure, a given...

Question 9

There must be equal amounts of mass on both sides of the center of mass (or center of gravity) of a system.

Accepted Answer

The center of mass (or center of gravity) is the point where the distribution of mass is balanced in all directions, but this does not require equal amounts of mass on both sides. It depends on the distribution of mass relative to that point.

Question 10

A car is traveling along a freeway at 65 mph. What is the linear speed, relative to the highway, of each of the following points on one of its tires?
(a) the highest point on the tire
(b) the lowest point on a tire
(c) the center of the tire

Accepted Answer

The answer of A car is traveling along a freeway...

Question 11

Two forces produce equal torques on a door about the door hinge. The first force is applied at the midpoint of the door; the second force is applied at the doorknob. Both forces are applied perpendicular to the door. Which force has a greater magnitude?
A) the first force (at the midpoint)
B) the second force (at the doorknob)
C) The two forces are equal.

Accepted Answer

The answer of Two forces produce equal torques on a...

Question 12

Two children, Ahmed and Jacques, ride on a merry-go-round. Ahmed is at a greater distance from the axis of rotation than Jacques. Which of the following are true statements? (There could be more than one correct choice.)
A) Jacques has a greater angular speed than Ahmed.
B) Jacques and Ahmed have the same angular speed.
C) Jacques has a smaller angular speed than Ahmed.
D) Ahmed has a greater tangential speed than Jacques.
E) Jacques and Ahmed have the same tangential speed.

Accepted Answer

B) When two points (in this case, Ahmed and Jacques) are on the same rotating object (the merry-go-round), they share the same angular speed because they complete a rotation in the same amount of time, regardless of their distance from the axis of rotation.D) Ahmed, being at a greater distance from the axis of rotation, has a greater tangential speed than Jacques. Tangential speed increases with distance from the axis of rotation, given by the formula $v = r\omega$, where $v$ is the tangential speed, $r$ is the radius (distance from the axis), and $\omega$ is the angular speed.

Question 13

The rotating systems shown in the figure differ only in that the two identical movable masses are positioned a distance r from the axis of rotation (left), or a distance r/2 from the axis of rotation (right). If you release the hanging blocks simultaneously from rest, 
A) the block at the left lands first.
B) the block at the right lands first.
C) both blocks land at the same time.

Accepted Answer

The answer of The rotating systems shown in the figure...

Question 14

Three small masses are positioned at the following coordinates: 3.0 kg at (3.0 m, 2.0 m); 4.0 kg at (0.0 m, -1.0 m); and 5.0 kg at (5.0 m, -7.0 m). What are the coordinates of the center of mass (or center of gravity) of this system?

Accepted Answer

The answer of Three small masses are positioned at the...

Question 15

The center of mass (or center of gravity) of a two-particle system is at the origin. One particle is located at (3.0 m, 0.0 m) and has a mass of 2.0 kg. The other particle has a mass of 3.0 kg. What is the location of the 3.0-kg particle?
A) (-3.0 m, 0.0 m)
B) (-2.0 m, 0.0 m)
C) (2.0 m, 0.0 m)
D) (3.0 m, 0.0 m)

Accepted Answer

The center of mass $x_{cm}$ of a system of particles is given by the formula $x_{cm} = \frac{\sum m_ix_i}{\sum m_i}$, where $m_i$ and $x_i$ are the masses and positions of the particles, respectively. Given that the center of mass is at the origin (0,0) and one particle is at (3.0 m, 0.0 m) with a mass of 2.0 kg, we can set up the equation $0 = \frac{(2.0\,kg \cdot 3.0\,m) + (3.0\,kg \cdot x)}{2.0\,kg + 3.0\,kg}$. Solving for $x$ gives $x = -2.0\,m$, indicating the 3.0-kg particle is at (-2.0 m, 0.0 m).

Question 16

A 30-kg child stands at one end of a floating 20-kg canoe that is 5.0-m long and initially at rest in the water. The child then slowly walks to the other end of the canoe. How far does the canoe move in the water, assuming water friction is negligible?
A) 1.0 m
B) 2.0 m
C) 3.0 m
D) 4.0 m
E) 5.0 m

Accepted Answer

According to the principle of conservation of momentum, when the child starts moving towards the other end, the total momentum of the system (child and canoe) must remain constant. Since the initial momentum is zero, the final momentum must also be zero. This means that the canoe must move in the opposite direction to compensate for the child's motion.

The child moves from one end to the other end, covering a distance of 5.0 m. Since the canoe is initially at rest, the center of mass of the system (child and canoe) is at rest. When the child moves to the other end, the center of mass of the system is displaced by a distance of 2.5 m (half the length of the canoe), towards the end where the child started. In order to maintain the center of mass of the system at rest, the canoe must move 2.5 m in the opposite direction.

Therefore, the canoe moves a total distance of 2.5 m + 0.5 m (the initial distance between the child and the end of the canoe) = 3.0 m in the water.

Hence, the correct choice is C) 3.0 m.

Question 17

As shown in the figure, a 60-cm length of uniform wire, of mass 60 g, is bent into a right triangle. What are the x and y coordinates of the center of mass (or center of gravity) of this triangle? 
A) (8.0 cm, 3.0 cm)
B) (8.0 cm, 5.0 cm)
C) (9.0 cm, 4.0 cm)
D) (10 cm, 3.0 cm)
E) (10 cm, 5.0 cm)

Accepted Answer

The answer of As shown in the figure, a 60-cm...

Question 18

A 60.0-kg man stands at one end of a 20.0-kg uniform 10.0-m long board. How far from the man is the center of mass (or center of gravity) of the man-board system?
A) 1.25 m
B) 2.50 m
C) 5.00 m
D) 7.50 m
E) 9.00 m

Accepted Answer

The answer of A 60.0-kg man stands at one end...

Question 19

Two equal-magnitude forces are applied to a door at the doorknob. The first force is applied perpendicular to the door, and the second force is applied at 30° to the plane of the door. Which force exerts the greater torque about the door hinge?
A) the first force (applied perpendicular to the door)
B) the second force (applied at an angle)
C) Both forces exert equal non-zero torques.
D) Both forces exert zero torque.

Accepted Answer

The torque exerted by a force depends on the angle at which the force is applied. The torque is maximized when the force is applied perpendicular to the lever arm (in this case, the door), as in the first force. The second force, applied at an angle, has a component perpendicular to the door that is less than the magnitude of the first force, resulting in a smaller torque.

Question 20

Three balls are moving along a straight line having the instantaneous positions shown in the figure. At that instant, find the location and velocity of the center of mass (or center of gravity) of this system.

Accepted Answer

The answer of Three balls are moving along a straight...

Quiz 7: Rotational Motion

Five forces act on a rod that is free to pivot at point P, as shown in the figure. Which of these forces is producing a counter-clockwise torque about point P? (There could be more than one correct choice.)

Three masses are located in the x-y plane as follows: a mass of 6 kg is at (0 m, 0 m) , a mass of 4 kg is at (3 m, 0 m) , and a mass of 2 kg is at (0 m, 3 m) . Where is the center of mass (or center of gravity) of the system?

In the figure, four point masses are placed as shown. Assume that all the numbers in the figure are accurate to two significant figures. What are the x and y coordinates of the center of mass (or center of gravity) of this arrangement?

A 2.0-m rope is lying on a table. You pick up one end and start raising it vertically. How high above the table is the center of mass (or center of gravity) of the rope when half of the rope has lifted off the table?

Three small masses are positioned as follows: 2.0 kg at (0.0 m, 0.0 m) , 2.0 kg at (2.0 m, 0.0 m) , and 4.0 kg at (2.0 m, 1.0 m) . Determine the coordinates of the center of mass (or center of gravity) of this system.

When a rigid object rotates about a fixed axis, what is true about all the points in the object? (There could be more than one correct choice.)

The figure shows scale drawings of four objects, each of the same mass and uniform thickness, with the mass distributed uniformly. Which one has the greatest moment of inertia when rotated about an axis perpendicular to the plane of the drawing at point P?

As shown in the figure, a given force is applied to a rod in several different ways. In which case is the torque about the pivot P due to this force the greatest? 1. 2. 3. 4. 5.

There must be equal amounts of mass on both sides of the center of mass (or center of gravity) of a system.

A car is traveling along a freeway at 65 mph. What is the linear speed, relative to the highway, of each of the following points on one of its tires? (a) the highest point on the tire (b) the lowest point on a tire (c) the center of the tire

Two forces produce equal torques on a door about the door hinge. The first force is applied at the midpoint of the door; the second force is applied at the doorknob. Both forces are applied perpendicular to the door. Which force has a greater magnitude?

Two children, Ahmed and Jacques, ride on a merry-go-round. Ahmed is at a greater distance from the axis of rotation than Jacques. Which of the following are true statements? (There could be more than one correct choice.)

The rotating systems shown in the figure differ only in that the two identical movable masses are positioned a distance r from the axis of rotation (left) , or a distance r/2 from the axis of rotation (right) . If you release the hanging blocks simultaneously from rest,

Three small masses are positioned at the following coordinates: 3.0 kg at (3.0 m, 2.0 m); 4.0 kg at (0.0 m, -1.0 m); and 5.0 kg at (5.0 m, -7.0 m). What are the coordinates of the center of mass (or center of gravity) of this system?

The center of mass (or center of gravity) of a two-particle system is at the origin. One particle is located at (3.0 m, 0.0 m) and has a mass of 2.0 kg. The other particle has a mass of 3.0 kg. What is the location of the 3.0-kg particle?

A 30-kg child stands at one end of a floating 20-kg canoe that is 5.0-m long and initially at rest in the water. The child then slowly walks to the other end of the canoe. How far does the canoe move in the water, assuming water friction is negligible?

As shown in the figure, a 60-cm length of uniform wire, of mass 60 g, is bent into a right triangle. What are the x and y coordinates of the center of mass (or center of gravity) of this triangle?

A 60.0-kg man stands at one end of a 20.0-kg uniform 10.0-m long board. How far from the man is the center of mass (or center of gravity) of the man-board system?

Two equal-magnitude forces are applied to a door at the doorknob. The first force is applied perpendicular to the door, and the second force is applied at 30° to the plane of the door. Which force exerts the greater torque about the door hinge?

Three balls are moving along a straight line having the instantaneous positions shown in the figure. At that instant, find the location and velocity of the center of mass (or center of gravity) of this system.