Question 1

When a 2.75-kg fan, having blades 18.5 cm long, is turned off, its angular speed decreases uniformly from 10.0 rad/s to 6.30 rad/s in 5.00 s.
(a) What is the magnitude of the angular acceleration of the fan?
(b) Through what angle (in degrees) does it turn while it is slowing down during the 5.00 s?
(c) If its angular acceleration does not change, how long after it is turned off does it take the fan to stop.

Accepted Answer

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Question 2

A 1.15-kg grinding wheel 22.0 cm in diameter is spinning counterclockwise at a rate of 20.0 revolutions per second. When the power to the grinder is turned off, the grinding wheel slows with constant angular acceleration and takes 80.0 s to come to a rest. (a) What was the angular acceleration (in rad/s²) of the grinding wheel as it came to rest if we take a counterclockwise rotation as positive? (b) How many revolutions did the wheel make during the time it was coming to rest?

Accepted Answer

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Question 3

A 2.00-m rod of negligible mass connects two very small objects at its ends. The mass of one object is 1.00 kg and the mass of the other is unknown. The center of mass of this system is on the rod a distance 1.60 m from the 1.00-kg mass object. What is the mass of the other object?
A) 4.11 kg
B) 3.22 kg
C) 4.00 kg
D) 0.250 kg
E) 0.800 kg

Accepted Answer

Let the mass of the unknown object be $m$. Then, the center of mass of the system is given by: $$\frac{(1.00	ext{ kg})(1.60	ext{ m}) + m(0.40	ext{ m})}{1.00	ext{ kg}+m} = 2.00	ext{ m}\left(\frac{1}{2}ight) = 1.00	ext{ m}$$ Simplifying this equation gives: $$1.60 + 0.40\frac{m}{1.00	ext{ kg}+m} = 1.00$$ Multiplying through by $1.00	ext{ kg}+m$ and rearranging gives: $$0.40m = 0.40	ext{ kg}$$ Hence, $m = \boxed{	extbf{(C) }4.00	ext{ kg}}$.

Question 4

A 3.45-kg centrifuge takes 100 s to spin up from rest to its final angular speed with constant angular acceleration. A point located 8.00 cm from the axis of rotation of the centrifuge moves with a speed of 150 m/s when the centrifuge is at full speed. (a) What is the angular acceleration (in rad/s²) of the centrifuge as it spins up? (b) How many revolutions does the centrifuge make as it goes from rest to its final angular speed?

Accepted Answer

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Question 5

A uniform piece of wire, 20 cm long, is bent in a right angle in the center to give it an L-shape. How far from the bend is the center of mass of the bent wire?
A) 2.5 cm
B) 3.5 cm
C) 4.5 cm
D) 5.0 cm
E) 7.1 cm

Accepted Answer

The center of mass of each segment of the wire (10 cm long) is at its midpoint, 5 cm from the bend. Since the wire is uniform and the segments are equal in length, the center of mass of the L-shape will be located at the point where the medians of the right triangle (formed by the two segments and the line connecting their midpoints) intersect. This point is 1/3 of the way along the median from the right angle, which is $\sqrt{5^2 + 5^2} = \sqrt{50}$ cm long. Therefore, the distance from the bend to the center of mass is $\frac{1}{3} \times \sqrt{50} \approx 3.5$ cm.

Question 6

If  = -4  - 2  - 3  , what is  ×  ?
A) +3  - 4 
B) +3  + 4 
C) -3  + 4 
D) +3  + 2  - 4 
E) -3  - 2  + 4

Accepted Answer

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Question 7

A 4.50-kg wheel that is 34.5 cm in diameter rotates through an angle of 13.8 rad as it slows down uniformly from 22.0 rad/s to 13.5 rad/s. What is the magnitude of the angular acceleration of the wheel? A) 0.616 rad/s² B) 5.45 rad/s² C) 111 rad/s² D) 22.5 rad/s² E) 10.9 rad/s²

Accepted Answer

The answer of A 4.50-kg wheel that is 34.5 cm...

Question 8

What is the magnitude of the cross product of a vector of magnitude 2.00 m pointing east and a vector of magnitude 4.00 m pointing 30.0° west of north?
A) 6.93
B) -6.93
C) 4.00
D) -4.00
E) 8.00

Accepted Answer

The magnitude of the cross product of two vectors is given by |A x B| = |A||B|sin(θ), where |A| and |B| are the magnitudes of the vectors, and θ is the angle between them. Here, the angle between the eastward vector and the vector pointing 30° west of north (which is 60° east of north) is 90° + 30° = 120°. Thus, |A x B| = 2 * 4 * sin(120°) = 8 * (√3/2) = 4√3 ≈ 6.93.

Question 9

What is the vector product of  = 2.00  + 3.00  + 1.00  and  = 1.00  - 3.00  - 2.00  ?
A) -3.00  + 5.00  - 9.00 
B) -5.00  + 2.00  - 6.00 
C) -9.00  - 3.00  - 3.00 
D) -4.00  + 3.00  - 1.00 
E) 2.00  -9.00  - 2.00

Accepted Answer

The answer of What is the vector product of ...

Question 10

A turbine blade rotates with angular velocity ω(t) = 2.00 rad/s- 2.1.00 rad/s³ . What is the angular acceleration of the blade at A) -38.2 rad/ B) -19.1 rad/ C) -86.0 rad/ D) -36.2 rad/ E) -172 rad/

Accepted Answer

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Question 11

If the magnitude of the cross product of two vectors is one-half the dot product of the same vectors, what is the angle between the two vectors?

Accepted Answer

The answer of If the magnitude of the cross product...

Question 12

In the figure, a 60-cm length of uniform wire, of 60 g mass and negligible thickness, is bent into a right triangle. The x and y coordinates of the center of mass, in cm, are closest to 
A) (8, 3).
B) (8, 5).
C) (9, 4).
D) (10, 3).
E) (10, 5).

Accepted Answer

The answer of In the figure, a 60-cm length of...

Question 13

For the vectors shown in the figure, find the magnitude and direction of  ×  , assuming that the quantities shown are accurate to two significant figures. 
A) 26, directed into the plane
B) 26, directed out of the plane
C) 31, directed on the plane
D) 31, directed into the plane
E) 31, directed out of the plane

Accepted Answer

The answer of For the vectors shown in the figure,...

Question 14

In the figure, four point masses are placed as shown. The x and y coordinates of the center of mass are closest to 
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

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Question 15

For the vectors shown in the figure, find the magnitude and direction of the vector product  ×  , assuming that the quantities shown are accurate to two significant figures. 
A) 16, directed into the plane
B) 16, directed out of the plane
C) 45, directed on the plane
D) 45, directed into the plane
E) 45, directed out of the plane

Accepted Answer

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Question 16

A heavy boy and a lightweight girl are balanced on a massless seesaw. If they both move forward so that they are one-half their original distance from the pivot point, what will happen to the seesaw? Assume that both people are small enough compared to the length of the seesaw to be thought of as point masses.
A) It is impossible to say without knowing the masses.
B) It is impossible to say without knowing the distances.
C) The side the boy is sitting on will tilt downward.
D) Nothing will happen; the seesaw will still be balanced.
E) The side the girl is sitting on will tilt downward.

Accepted Answer

The seesaw will remain balanced because both individuals move closer to the pivot point by the same proportion, maintaining the balance due to the principle of moments (torque). The product of their masses and distances from the pivot (mass x distance) remains constant for both, preserving equilibrium.

Question 17

The angular acceleration of a wheel is given in  by  where t is in seconds. If the wheel starts from rest at  when is the next time the wheel is at rest?
A) 5.1 s
B) 8.4 s
C) 6.9 s
D) 3.6 s

Accepted Answer

The answer of The angular acceleration of a wheel is...

Question 18

The angular velocity of a 755-g wheel 15.0 cm in diameter is given by the equation ω(t) = (2.00 rad/s²)t + (1.00 rad/s⁴)t³. (a) Through how many radians does the wheel turn during the first 2.00 s of its motion? (b) What is the angular acceleration (in rad/s²) of the wheel at the end of the first 2.00 s of its motion?

Accepted Answer

The answer of The angular velocity of a 755-g wheel...

Question 19

A long thin rod of length L has a linear density λ(x) = Ax where x is the distance from the left end of the rod.
(a) How far is the center of mass of the rod from the left end of the rod?
(b) What is the mass of the rod?

Accepted Answer

The answer of A long thin rod of length L...

Question 20

If  = -2  - 6  + 2  and  = -2  -2  - 3  , which of the following numbers is closest to the magnitude of  ×  ?
A) 25
B) 21
C) 17
D) 13
E) 9

Accepted Answer

The answer of If  = -2  - 6...

Quiz 12: Rotation of a Rigid Body

A 2.00-m rod of negligible mass connects two very small objects at its ends. The mass of one object is 1.00 kg and the mass of the other is unknown. The center of mass of this system is on the rod a distance 1.60 m from the 1.00-kg mass object. What is the mass of the other object?

A uniform piece of wire, 20 cm long, is bent in a right angle in the center to give it an L-shape. How far from the bend is the center of mass of the bent wire?

If = -4 - 2 - 3 , what is × ?

A 4.50-kg wheel that is 34.5 cm in diameter rotates through an angle of 13.8 rad as it slows down uniformly from 22.0 rad/s to 13.5 rad/s. What is the magnitude of the angular acceleration of the wheel?

What is the magnitude of the cross product of a vector of magnitude 2.00 m pointing east and a vector of magnitude 4.00 m pointing 30.0° west of north?

What is the vector product of = 2.00 + 3.00 + 1.00 and = 1.00 - 3.00 - 2.00 ?

A turbine blade rotates with angular velocity ω(t) = 2.00 rad/s- 2.1.00 rad/s3 . What is the angular acceleration of the blade at

If the magnitude of the cross product of two vectors is one-half the dot product of the same vectors, what is the angle between the two vectors?

In the figure, a 60-cm length of uniform wire, of 60 g mass and negligible thickness, is bent into a right triangle. The x and y coordinates of the center of mass, in cm, are closest to

For the vectors shown in the figure, find the magnitude and direction of × , assuming that the quantities shown are accurate to two significant figures.

In the figure, four point masses are placed as shown. The x and y coordinates of the center of mass are closest to

For the vectors shown in the figure, find the magnitude and direction of the vector product × , assuming that the quantities shown are accurate to two significant figures.

The angular acceleration of a wheel is given in by where t is in seconds. If the wheel starts from rest at when is the next time the wheel is at rest?

A long thin rod of length L has a linear density λ(x) = Ax where x is the distance from the left end of the rod. (a) How far is the center of mass of the rod from the left end of the rod? (b) What is the mass of the rod?

If = -2 - 6 + 2 and = -2 -2 - 3 , which of the following numbers is closest to the magnitude of × ?

A turbine blade rotates with angular velocity ω(t) = 2.00 rad/s- 2.1.00 rad/s³ . What is the angular acceleration of the blade at