ROTATIONAL KINEMATICS
CHAPTER 8
CONCEPTUAL QUESTIONS
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1.
REASONING AND SOLUTION The figures below show two axes in the plane of the
paper and located so that the points B and C move in circular paths having the same radii
(radius = r).
C
r
A
r
Axis
B
C
r
Axis
A
B
r
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2.
REASONING AND SOLUTION When a pair of scissors is used to cut a string, each blade
of the scissors does not have the same angular velocity at a given instant during the cut. The
angular speed of each blade is the same; however, each blade rotates in the opposite
direction. Therefore, it is correct to conclude that the blades have opposite angular velocities
at any instant during the cut.
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ROTATIONAL KINEMATICS
3.
REASONING AND SOLUTION Just before the clock is unplugged, the second hand is
rotating so that its angular velocity is clockwise. The second hand moves with a constant
angular velocity, so the angular acceleration is zero. When the plug is pulled, the second
hand will continue to rotate with its clockwise angular velocity, but it will slow down.
Therefore, the angular acceleration must be opposite to the angular velocity, or
counterclockwise.
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4.
REASONING AND SOLUTION The tangential speed, vT, of a point on the earth's surface
is related to the earth's angular speed  according to vT  r , Equation 8.9, where r is the
perpendicular distance from the point to the earth's rotation axis. At the equator, r is equal
to the earth's radius. As one moves away from the equator toward the north or south
geographic pole, the distance r becomes smaller. Since the earth's rotation axis passes
through the geographic poles, r is effectively zero at those locations. Therefore, your
tangential speed would be a minimum if you stood as close as possible to either the north or
south geographic pole.
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5.
REASONING AND SOLUTION
a. A thin rod rotates at a constant angular speed about an axis of rotation that is
perpendicular to the rod at its center. As the rod rotates, each point at a distance r from the
center on one half of the rod has the same tangential speed as the point at a distance r from
the center on the other half of the rod. This is true for all values of r for 0  r  (L/ 2)
where L is the length of the rod.
b. If the rod rotates about an axis that is perpendicular to the rod at one end, no two points
are the same distance from the axis of rotation. Therefore, no two points on the rod have the
same tangential speed.
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6.
REASONING AND SOLUTION The wheels are rotating with a constant angular velocity.
a. Since the angular velocity is constant, each wheel has zero angular acceleration,  =
0 rad/s. Since the tangential acceleration aT is related to the angular acceleration through
Equation 8.10, aT  r , every point on the rim has zero tangential acceleration.
b. Since the particles on the rim of the wheels are moving along a circular path, they must
have a centripetal acceleration. This can be supported by Equation 8.11, ac  r 2 , where
ac is the magnitude of the centripetal acceleration. Since  is nonzero, ac is nonzero.
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Chapter 8 Conceptual Questions
7.
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REASONING AND SOLUTION
The wheel
and the two points under consideration are shown
in the figure at the right.
a. Each point must undergo the same angular
displacement in the same time interval.
Therefore, at any given instant, the angular
velocity of both points is the same.
1
2
C
b. Each point must increase its angular velocity
at the same rate; therefore, each point has the
same angular acceleration.
c. The tangential speed is given by vT  r . Since both points have the same angular speed
, but point 1 is further from the center than point 2, point 1 has the larger tangential speed.
d. The tangential acceleration is given by aT  r . Since both points have the same angular
acceleration , but point 1 is further from the center than point 2, point 1 has the larger
tangential acceleration.
e. The centripetal acceleration is given by ac  r 2 . Since both points have the same
angular speed , but point 1 is further from the center than point 2, point 1 has the larger
centripetal acceleration.
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8.
REASONING AND SOLUTION For a building that is located on the earth's equator, the
points in the building rotate about the center of the earth. The tangential speed of any point
is given by vT  r , where, in this case, r is measured from the center of the earth. The top
floor of the building has a larger value of r than the other floors; therefore, the top floor has
the greater tangential speed.
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9.
REASONING AND SOLUTION The centripetal acceleration of any point in the space
station is given by ac  r 2 (Equation 8.11),where r is the distance from the point to the
axis of rotation of the space station. If the centripetal acceleration is adjusted to be g at a
given value of r (such as the astronaut's feet), the centripetal acceleration will be different at
other values of r. Therefore, if the adjustment is made so that the centripetal acceleration at
the astronaut's feet equals g, then the centripetal acceleration will not equal g at the
astronaut's head.
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10. REASONING AND SOLUTION The tangential speed for any point on the tip of a clock
hand is given by vT  r . The angular speeds of the second hand, the minute hand, and the
hour hand differ greatly, with the second hand having the largest angular speed and the hour
hand having the smallest. If one desired to create a clock in which the tips of the second
hand, the minute hand, and the hour hand moved with the same tangential speed, the lengths
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ROTATIONAL KINEMATICS
of the arms would also have to differ greatly, with the second hand having the smallest arm
and the hour hand having the largest. Such a clock would not be very practical.
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11. REASONING AND SOLUTION Any point on a rotating object possesses a centripetal
acceleration that is directed radially toward the axis of rotation. This also applies to a tire on
a moving car and is true regardless of whether the car has a constant linear velocity or
whether it is accelerating.
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12. REASONING AND SOLUTION The bicycle wheel has an angular acceleration. The
arrows are perpendicular to the radius of the wheel. The magnitude of the arrows increases
with increasing distance from the center in accordance with vT  r , or aT  r . The
arrows in the picture could represent either the tangential velocity or the tangential
acceleration. The arrows are not directed radially inward; therefore, they cannot represent
the centripetal acceleration.
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13. REASONING AND SOLUTION The speedometer of a truck uses a device that measures
the angular speed of the tires. The angular speed is related to the linear speed of the truck by
v  r .
Suppose two trucks are traveling side-by-side along a highway at the same linear speed v,
and one truck has larger wheels than the other. The angular speed of the larger-diameter
wheels is less than that of the smaller-diameter wheels. Since the speedometer uses a device
that measures the angular speed of the tires, the speedometer reading on the truck with the
larger wheels will indicate a smaller linear speed than that on the truck with the smaller
wheels.
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14. REASONING AND SOLUTION When a fan is shut off, the fan blades gradually slow
down until they eventually stop. The angular acceleration is never really constant because it
gradually decreases, becoming zero at the instant the blades stop rotating. In solving
problems, it is sometimes convenient to approximate such motion as having constant angular
acceleration. This is a fairly good approximation for a short time interval just after the
switch is turned off.
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15. REASONING AND SOLUTION Three examples that involve rotation about an axis that is
not fixed:
(1) The motion of a Frisbee. The axis of the rotating Frisbee moves through the air
with the Frisbee.
(2) The motion of the earth in its orbit. The rotation axis of the earth changes
position as the earth revolves around the sun.
(3) The motion of a twirling baton that has been thrown into the air. The rotation axis
constantly changes location as the baton rises and falls.
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