Deck 12: Rotation of a Rigid Body

Question 1

The mathematical relationship between the angular acceleration and the angular velocity of a body is analogous to the mathematical relationship between

A) the angular velocity and the angle.
B) the angle and the angular velocity.
C) the angular velocity and the time.
D) the angle and the time.

Answer 1

the angular velocity and the angle.

Question 2

Dimensionally, correct relations among angle $\theta$ , angular velocity $\omega$ , and angular acceleration $\alpha$ include all of the following except

A) $[ \alpha ] = [ \omega ] \cdot [ \theta ]$
B) $[ \omega ] = [ \alpha ] \cdot [ \mathrm { t } ]$
C) $[ \theta ] = [ \alpha ] \cdot \left[ \mathrm { t } ^ { 2 } \right]$
D) $[ \omega ] = [ \alpha ] \cdot [ \theta ]$

Answer 2

$[ \alpha ] = [ \omega ] \cdot [ \theta ]$

Question 3

For an object undergoing constant angular acceleration, all of the following change with time except

A) the slope of a graph of angular velocity versus time.
B) the angular position of the object.
C) the average angular velocity of the object.
D) the instantaneous angular velocity of the object.

Answer 3

the slope of a graph of angular velocity versus time.

Question 4

An object rotates with a constant angular acceleration of 10 rad/s², an initial angular speed of 10 rad/s, and a final angular speed of 30 rad/s. Its total time of rotation is

A) 1.0 s.
B) 2.0 s.
C) 3.0 s.
D) 4.0 s.

Question 5

An object rotates with a constant angular acceleration of 10 rad/s², an initial angular speed of 10 rad/s, and a final angular speed of 30 rad/s. Its average angular speed is

A) 10 rad/s.
B) 20 rad/s.
C) 30 rad/s.
D) 40 rad/s.

Question 6

An object rotates with a constant angular acceleration of 10 rad/s², an initial angular speed of 10 rad/s, and a final angular speed of 30 rad/s. The angle turned through is

A) 10 rad.
B) 20 rad.
C) 30 rad.
D) 40 rad.

Question 7

An object rotates with a constant angular acceleration of 5.0 rad/s², an initial angular speed of -10 rad/s, and a total time of rotation of 4.0 seconds. The average angular velocity has a magnitude of

A) +10 rad/s.
B) +20 rad/s.
C) -10 rad/s.
D) zero.

Question 8

An object rotates with a constant angular acceleration of 5.0 rad/s², an initial angular speed of -10. rad/s, and a total time of rotation of 4.0 s. The instant at which the object has a zero instantaneous angular speed is

A) 1.0 s.
B) 2.0 s.
C) 3.0 s.
D) 4.0 s.

Question 9

The dimensions of the moment of inertia, I, are the same as those of

A) [mass].
B) [mass]·[distance].
C) [mass]·[distance]².
D) [mass]/[distance]².

Question 10

A point on a rotating object at a distance $R$ from the axis of rotation has associated with it several angular kinematic variables: namely $\theta , \omega$ , and $\alpha$ . To obtain the corresponding linear variables, one can multiply the respective angular variable by the factor

A) 1/R².
B) 1/R.
C) R.
D) R².

Question 11

The relation between the angular velocity $\omega$ and the period T of a rotating object is

A) $\omega$
= 2 $\pi$ T.
B) T = 2 $\pi$ $\omega$
C) 2 $\pi$ $\omega$ = l/T.
D) T = 2 $\pi$ / $\omega$

Question 12

A hollow sphere, a solid sphere, a hollow right cylinder (or hoop), and a solid right cylinder, each with the same total mass and identical maximum radius, are each rotated about their axis of symmetry. The object with the largest moment of inertia is the

A) hollow sphere.
B) solid sphere.
C) hollow right cylinder (or hoop).
D) solid right cylinder.

Question 13

A particle moves with constant velocity. At instant t = 0, the particle is at the position of minimum separation from fixed point P. Its angular momentum about P is L ( $\not \neq$ 0). Subsequently its angular momentum about point P

A) increases with time.
B) decreases with time.
C) does not change with time.
D) Hold it! A particle moving with constant velocity has no angular momentum.

Question 14

A wheel with an initial $\omega = 0$ rotates at constant $\alpha ( \neq 0 )$ . A while later, the acceleration of a point on the rim must include all of the following except

A) a tangential component.
B) a radial component.
C) a fixed direction, with respect to a radial line connecting the point on the rim to the axis of rotation.
D) Hold it! There are no exceptions.

Question 15

The centripetal acceleration for a point on a rotating object is

A) R $\omega$
B) R² $\omega$
C) $\omega$ /R².
D) R $\omega$ ² .

Question 16

A flat plate is placed within the plane of the paper. The plate is rotating counterclockwise around an axis that is perpendicular to the plate. The direction of its angular momentum is

A) out of the paper toward the observer.
B) into the paper away from the observer.
C) counterclockwise in the plane of the paper.
D) clockwise in the plane of the paper.

Question 17

In many situations in physics, the order of two events does not affect the final outcome. An exception to this rule occurs for the case of

A) two successive rotations performed on a body about different axes.
B) two successive translations given to a body about different axes.
C) two successive rotations performed on a body about the same axis.
D) two successive translations given to a body about the same axis.

Question 18

A thin rod of length L has a mass/length ratio proportional to x-the distance measured from end A. The moment of inertia of the rod about an axis passing through A, perpendicular to the rod, is

A) proportional to x.
B) proportional to x².
C) proportional to x³.
D) proportional to x⁴.

Question 19

The Earth on its path around the Sun has an orbital angular momentum J and a spin angular momentum S due to its rotation about its own axis. Knowing the relationship between the year and the day, the Earth's radius, and the Earth-Sun distance, the ballpark relation between the magnitudes of J and S is

A) J larger than S.
B) J approximately equal to S.
C) J less than S.
D) Hold it! Comparing these two quantities is not legitimate because they are quantities that do not have anything in common.

Question 20

The direction of the angular momentum for an extended body lies along the axis of rotation

A) if that axis is an axis of symmetry.
B) for small rotational speeds.
C) both of the above answers are correct.
D) neither of the above answers is correct.

Question 21

A uniform, flat, square plate has a mass of 3.0 kg and a side 2.0 m in length. The moment of inertia about an axis that passes through its center and is parallel to one edge is 1.0 kg·m². The moment of inertia of the plate about one edge is

A) 2.0 kg·m².
B) 4.0 kg·m².
C) 12 kg·m².
D) 13 kg·m².

Question 22

A uniform, flat, square plate has a mass of 3.0 kg and a side 2.0 m in length. The moment of inertia about an axis that passes through its center and is parallel to one edge is 1.0 kg·m². The moment of inertia of the plate about an axis perpendicular to the plane through its center is

A) 2.0 kg·m².
B) 4.0 kg·m².
C) 12 kg·m².
D) 13 kg·m².

Question 23

Two masses, M₁ = 2.0 kg and M₂ = 4.0 kg, are joined by a rigid rod of negligible mass and length 6.0 m. The moment of inertia of the system about an axis perpendicular to the rod and passing through the center of mass is

A) 12 kg·m².
B) 24 kg·m².
C) 36 kg·m².
D) 48 kg·m².

Question 24

The average distance between the Earth and the Sun is 1.5 $\times$ 10¹¹ m. The tangential speed of the Earth in its orbit about the Sun is

A) 45 km/s.
B) 60 km/s.
C) 15 km/s.
D) 30 km/s.

Question 25

The dimensions of the radian are

A) length/degrees.
B) degrees/length.
C) length.
D) Hold it! The radian is dimensionless.

Question 26

An object undergoes a rotation of 115° in a circular path 1.0 m in radius. The length of circular arc the object traverses is

A) 1.0 $\times$ 10² m.
B) 2.0 m.
C) 1.0 m.
D) 0.0 m.

Question 27

The average Earth-Sun distance is approximately 1.5 x 10¹¹ m, and the Sun's radius is approximately 7.0 x 10⁸ m. The approximate angle (in degrees) that the Sun subtends when viewed from Earth is

A) 1/4°.
B) 1/2°.
C) 1°.
D) 2°.

Question 28

A car's engine redlines (reaches its maximum rotational frequency) at 8000 rpm. The time taken for one revolution is

A) 2.5 ms.
B) 5.0 ms.
C) 7.5 ms.
D) 10. ms.

Question 29

A wheel undergoes rotational motion according to the equation $\phi ( \mathrm { rad } ) = c t + d t ^ { 3 }$ , where $\surd$ is in radians, c = 1.0 rad/s, and d = -0.50 rad/s³. The angle the wheel has turned in 2.0 s is

A) -2.0 rad.
B) 0 rad.
C) 2.0 rad.
D) 5.0 rad.

Question 30

A wheel undergoes rotational motion according to the equation $\phi ( \mathrm { rad } ) = c t + d t ^ { 3 }$ , where $\surd$ is in radians, c = 1.0 rad/s, and d = -0.50 rad/s³. The wheel's angular velocity at 2.0 s is

A) -7.0 rad/s.
B) 7.0 rad/s.
C) 5.0 rad/s.
D) -5.0 rad/s.

Question 31

A wheel undergoes rotational motion according to the equation $\phi ( \mathrm { rad } ) = c t + d t ^ { 3 }$ , where $\surd$ is in radians, c = 1.0 rad/s, and d = -0.50 rad/s³. The wheel's instantaneous angular acceleration at t = 2.0 s is

A) -6.0 rad/s².
B) -3 rad/s².
C) 0 rad/s².
D) 3 rad/s².

Question 32

A wheel undergoes rotational motion according to the equation $\phi ( \mathrm { rad } ) = c t + d t ^ { 3 }$ , where $\surd$ is in radians, c = 1.0 rad/s, and d = -0.50 rad/s³. At t = 2.0 s the magnitude of the tangential speed of an object placed 10.0 cm from the center of the wheel is

A) 1.0 m/s.
B) 0.50 m/s.
C) 0.0 m/s.
D) 1.5 m/s.
E) 2.0 m/s.

Question 33

A wheel undergoes rotational motion according to the equation $\phi ( \mathrm { rad } ) = c t + d t ^ { 3 }$ , where $\surd$ is in radians, c = 1.0 rad/s, and d = -0.50 rad/s³. At t = 2.0 s the magnitude of the tangential acceleration of an object placed 10.0 cm from the center of the wheel is

A) -0.60 m/s².
B) -0.30 m/s².
C) 0 m/s².
D) 0.30 m/s².

Question 34

A wheel undergoes rotational motion according to the equation $\phi ( \mathrm { rad } ) = c t + d t ^ { 3 }$ , where $\surd$ is in radians, c = 1.0 rad/s, and d = -0.50 rad/s³. At t = 2.0 s the centripetal acceleration of an object 10.0 cm from the center of the wheel is

A) 5.0 m/s².
B) 2.5 m/s².
C) 1.3 m/s².
D) 0 m/s².

Question 35

A wheel undergoes rotational motion according to the equation $\phi ( \mathrm { rad } ) = c t + d t ^ { 3 }$ , where $\surd$ is in radians, c = 1.0 rad/s, and d = -0.50 rad/s³. At t = 2.0 s the net acceleration of an object 10.0 cm from the center of the wheel is

A) 0 m/s².
B) 1.3 m/s².
C) 2.6 m/s².
D) 3.9 m/s².

Question 36

A car can accelerate from zero to 25 m/s in 9.5 seconds. If the diameter of the car's tire is 28 cm, the angular acceleration of the tire is

A) 89 rad/s².
B) 19 rad/s².
C) 49 rad/s².
D) 9.4 rad/s².

Question 37

Two point masses of 5.0 kg and 15.0 kg, separated by 2.0 m, are spinning about their center of mass. The moment of inertia of the system is

A) 5.0 kg^.m².
B) 10 kg^.m².
C) 16 kg^.m².
D) 20 kg^.m².

Question 38

Two point masses of 5.0 kg and 15.0 kg, separated by 2.0 m, are spinning about their combined center of mass. When the system rotates at 0.80 rad/s, the kinetic energy for the motion is

A) 5.0 J.
B) 3.8 J.
C) 0 J.
D) 13 J.

Question 39

An object is rotating with an angular acceleration given by $\alpha = c + d t ^ { 2 }$ , where $\surd$ is in radians/s², t is in seconds, c = -3.0 rad/s², and d = 0.40 rad/s⁴. At t = 2.0 s the angular acceleration of the object is

A) -2.2 rad/s².
B) -1.4 rad/s².
C) 3.8 rad/s².
D) 4.6 rad/s².

Question 40

An object, starting from rest, rotates with an angular acceleration given by $\alpha = c + d t ^ { 2 }$ , where $\surd$ is in radians/s², t is in seconds, c = -3.0 rad/s² and d = 0.40 rad/s⁴. At t = 2.0 s the angular velocity of the object is

A) -1.1 rad/s.
B) 1.1 rad/s.
C) -4.9 rad/s.
D) -7.1 rad/s.

Question 41

An object, starting from rest at the origin, rotates with an angular acceleration given by $\alpha = c + d t ^ { 2 }$ , where $\surd$ is in radians/s², t is in seconds, c = -3.0 rad/s², and d = 0.40 rad/s⁴. At time t = 2.0 s, the rotation (in radians) the object has made is

A) -1.1 rad.
B) 1.1 rad.
C) -4.9 rad.
D) -5.5 rad.

Question 42

A thin rod 1.0 m in length has a mass of 1.0 kg. Moving the axis of rotation will change the moment of inertia of the rod. The position, relative to the end of the rod, of the axis of rotation perpendicular to the rod for the moment of inertia to be three times the moment of inertia about an axis perpendicular to the rod passing through its center of mass is

A) 5.3 cm.
B) 9.2 cm.
C) 11 cm.
D) 25 cm.

Question 43

A cylinder starting from rest is accelerated at 5.5 rad/s². The number of revolutions the cylinder turns through in 4.0 s is

A) 7.0.
B) 25.
C) 35.
D) 44.

Question 44

Two uniform solid spheres have masses M₀ and (1/2)M₀ and radii R₀ and (1/2)R₀ respectively. The ratio of their moments of inertia about their own diameters is

A) 8:1.
B) 4:1.
C) 2:1.
D) 1:1.

Question 45

Two masses, M_A = 1.0 kg and M_B = 2.5 kg, are attached at the opposite ends of a rod 1.0 m in length. The mass of the rod is 1.5 kg. The moment of inertia of the system about an axis perpendicular to the rod passing through its center is

A) 1.0 kg^.m².
B) 1.5 kg^.m².
C) 2.0 kg^.m².
D) 2.5 kg^.m².

Question 46

A uniform solid sphere with a mass of 0.15 kg rolls without slipping with a tangential velocity of 1.0 m/s. The kinetic energy of rotation of the sphere is

A) 30 mJ.
B) 15 mJ.
C) 60 mJ.
D) 45 mJ.

Question 47

An object is placed on a turntable (rotating platter) 25.0 cm from the center of rotation. The coefficient of friction between the turntable and object is 0.40. The turntable starts from rest (at time t = 0) with and is accelerated at 0.10 rad/s². The object will begin to slide off the turntable after

A) 20 s.
B) 40 s.
C) 60 s.
D) Not enough information is given to solve this problem.

Question 48

A uniform solid sphere has a mass M and radius R. Its moment of inertia about an axis tangent to the surface of the sphere is:

A) (2/5)MR².
B) MR².
C) (7/5)MR².
D) (5/2)MR².

Question 49

A billiard ball of mass 0.16 kg, radius 2.86 cm, and moment of inertia about its diameter of 5.23 $\times$ 10^-5 kg·m² moves on a table at 1.5 m/s. The rotational kinetic energy of the ball is

A) 0 J.
B) 0.072 J.
C) 0.14 J.
D) 0.25 J.

Question 50

A billiard ball of mass 0.16 kg, radius 2.86 cm, and moment of inertia about its diameter of 5.23 $\times$ 10^-5 kg·m² moves on a table at 1.5 m/s. The total kinetic energy of the ball is

A) 0.072 J.
B) 0.14 J.
C) 0.25 J.
D) 0.32 J.