Page 193 - Fisika Terapan for Engineers and Scientists

Answers to Checkups 393

Answers to Checkups

cally down. The tangential acceleration points perpendicular
Checkup 12.1 to a radius at any point; since the elevator accelerates upward,
1. The swinging door executes only rotational motion about its the tangential acceleration at the top of the wheel points hori-
(fixed) hinges. The motions of the wheel of a train and of the zontally toward the left.
propeller of an airplane involve both rotational and transla- 2. Yes to both. As long as there is no slipping, we have v/R
tional motion; the wheel and propeller rotate as the vehicle and a/R, so the behavior of an angular quantity is the
moves through space. same as the corresponding translational quantity.
2. Yes, the motion is describable as rotation about an axis and 3. (D) 20. For constant acceleration and starting from rest, the
2
1
simultaneous translational motion. The rotational motion is angular position is f t 2 . Since this is proportional to t ,
2
rotation about an axis through the end of the hammer; the the angular position will be four times greater in twice the
translational motion, however, is not along a parabolic path, time. Thus the total number of revolutions in the first two sec-
but involves more complicated looping motion (see Fig. 12.1). onds is 4 5 20.
3. An automobile exhibits roll motion when driving on a banked
surface; the auto is then tilted. Pitch motion can occur during Checkup 12.4
sudden braking, when the front of the auto dives downward. 1 2
1. Since the angular velocity is proportional to t , the angular
Turning motion occurs whenever the auto is being driven
position, which is the integral of the angular velocity over time
around a curve (compare Fig. 12.2). 1 2 1 3/2
[Eq. (12.23)], will be proportional to t t . Thus the
4. (D) Swinging door. The axis of rotation is through the hinges, angular position will be 4 3/2 8 times as large at t 4 s as it
along the edge of the door. was at t 1 s.
4
2. (D) t . If the angular acceleration increases in proportion to
Checkup 12.2 the time t, then the angular velocity dt increases in
2
1. The point P has the larger instantaneous speed (it travels through proportion to t . The centripetal acceleration is given by
2
2
a greater distance per unit time). Both points have the same a centripetal v R R, and so increases in proportion to the
instantaneous angular velocity and the same angular accelera- fourth power of the time.
tion (as do all points on the same rigid body). Hence the point
P has the larger tangential acceleration (a R) and also Checkup 12.5
tangential
2
the larger centripetal acceleration (a R).
centripetal 1. Since all of the mass M is at the same distance from the axis of
2
2. The radius R for circular motion is the perpendicular distance rotation, the moment of inertia is simply I MR .
from the axis of rotation, and so is equal to the Earth’s radius only
2. Rotation about an axis perpendicular to the rod through its end
at the equator, and is increasingly smaller as one moves toward
gives the largest moment of inertia, since more mass is located
the poles; at a pole, R is zero. All points have the same angular
at a greater distance from the axis of rotation. Rotation about
velocity , as for any rigid body.The velocity is not the same for
an axis along the rod must give the smallest moment of inertia,
all points; since v R, v is largest at the equator. All points do since in this case all of the mass is very close to the axis.
2
not have the same centripetal acceleration; since a R,
centripetal 3. About an axis along one edge or through its center parallel
the centripetal acceleration is largest at the equator.
to one edge, the distribution of mass (relative to the axis of
3. There is a centripetal acceleration; at the top of the arc, this is rotation) in each case is the same as for the corresponding rod
2
directed downward (a v /R). There is no tangential
centripetal (imagine viewing Fig. 12.19b from above, that is, along the
acceleration at the top (no forces act in this direction). Some
axis of rotation). Thus the moment of inertia of the square
distance beyond the highest point, there will be both a cen- 1 2
about an axis along one edge is I ML ; about an axis
3
tripetal acceleration (since the car still moves along an arc) and 1 2
through its center parallel to one edge, it is ML .
12
a tangential acceleration (since now a component of the gravi-
4. About an axis through the center, each particle is a
tational force is tangent to the path).
distance l 2 from the axis, and so the moment of inertia is
4. (D) Handle end; handle end. Since the rotation is about an 2 2 1 2
I m(l 2) m(l 2) ml . About an axis through one
2
axis through the center of mass (near the hammer head), the
particle, one particle is a distance l from the axis and the
end of the handle is furthest from the axis.Thus both the speed other is at zero distance, so I ml 0 ml . Since we
2
2
2
v R and the centripetal acceleration a R are
centripetal have shifted the axis by d l 2 in the second case, we indeed
largest at the end of the handle, since R is largest there (and 2 1 2 2 2
have I I CM Md ml (2m)(l 2) ml , so the
2
is a constant for all points on a rigid body).
parallel-axis theorem is satisfied (notice we must use the total
mass M 2m).
Checkup 12.3 2
5. (B) 2MR . Since the axis is shifted by a distance d R, the
2 2 2
1. The centripetal acceleration always points toward the center of parallel-axis theorem gives I I CM Md MR MR
2
curvature of the circular arc of the problem; here, this is verti- 2MR for rotation about a point on the hoop.