Page 34 - Fisika Terapan for Engineers and Scientists

234 CHAPTER 7 Work and Energy

4. In each case, the force is opposite to the displacement 3. The kinetic energy of the golf ball is largest at the beginning
(whether pushing against the front or pulling on the rear, the (and end, if we neglect air resistance) of the trajectory; at
force is rearward), and so negative work is done on the cart in higher points, the force of gravity has slowed the ball down.
both cases. The kinetic energy is smallest at thetop of the trajectory,
5. No.The tension provides a centripetal acceleration, which is per- where there is only a horizontal contribution to the speed
2
2
pendicular to the (tangential) motion, and thus does no work. (v 2v v ). The kinetic energy is not zero while the
y
x
ball is in the air (unless the ball was accidentally launched ver-
6. The work is positive in (b) and (c), where the angle between
tically; in that case, the kinetic energy would be zero at the top
the force and displacement is less than 90 ; the work is nega-
of the trajectory).
tive in (a), where the angle is greater than 90 . The work is
zero in (d), where the force is perpendicular to the displace- 4. No. For the work–energy theorem to apply, one must con-
ment. The work is largest when the force is most nearly paral- sider the net external force on the sled. If traveling at con-
lel to the displacement; for force vectors (and displacement stant velocity (zero acceleration), the total force must be zero
vectors) of equal magnitude, this occurs in (c). (the horse’s pull does positive work and is canceled by the
friction force, which does negative work), and so the total
7. (E) 4 and 5. To calculate the work done by a constant force, W
work done on the sled is zero. Thus there is no change in
Fs cos , you do not need to know the mass, acceleration, or
kinetic energy.
speed. You do need to know the force, the displacement, and 1 2
the angle between the two. 5. (E) 9. The kinetic energy, K mv , is proportional to the
2
square of the speed; thus increasing the speed by a factor of 3
Checkup 7.2 increases the kinetic energy by a factor of 9.
1. The work done by a variable force is equal to the area under the Checkup 7.4
F(x) vs. x curve. Assuming the two plots are drawn to the same
vertical scale, for a displacement from a to b, the upper plot 1. As in Example 8, the velocity at the bottom depends only on
clearly has a greater area between the F(x) curve and the x axis. the height of release (the cars do not even have to have the
same mass!); thus, the upper roller coaster will provide the
2. If we consider a plot such as Fig. 7.13 and imagine extending
larger speed at the bottom, since y is greater.
the curve to the left to x b [where F(x) kb], then we
see that positive work is done on the particle as it moves from 2. The gravitational potential energy U decreases as the piano is
x b to x 0 [where the area between the F(x) curve and brought to street level from the first house; U remains con-
the x axis is above the x axis]. Negative work is done on the stant during the trip to the nearby house (assuming travel
particle as it moves from x 0 to x b [where the area over flat ground); then, the gravitational potential energy
between the F(x) curve and the x axis is below the origin]. increases back to its original value as the piano is brought up
Thus the net work is zero. to the second floor of the second house (assuming similar
houses).
3. The work you must do on the spring is the opposite of what
the spring does on you, since the forces involved are an 3. No. At constant speed, K is constant; since U decreases as the
action–reaction pair. Thus the work you do is the negative of truck moves down, E K U decreases also, and so is not
1
2
2
the result of Example 4, or W k(b a ). conserved.
2
2
1
4. (D) 3W.The work to stretch from equilibrium is kx , so the first 4. Since both the girl and the boy change height by the same
2
1
2
stretch requires W kd .The second stretch requires work amount, they both reach the pool with the same speed (at any
2
1 2 2d 1 2 1 2 vertical height, they have the same speed, but the boy’s velocity
W kx ` k (2d) kd 4W W 3W.
2 d 2 2
has a horizontal component, so his vertical velocity is slower
Checkup 7.3 than that of the girl). Since the girl’s velocity is all vertical, a
larger vertical velocity implies that she reaches the pool first.
1
2
1. Yes—the kinetic energy, K mv , depends only on the
2
square of the speed, and not on the direction of the velocity. 5. (A) 22. As in Example 8, the speed at the bottom (starting
from rest) is proportional to the square root of the initial
Thus if the two equal masses have the same speed, they have
height. Thus, for twice the height, the speed of the first bicy-
the same kinetic energy.
clist will be 22 times as large at the bottom.
2. Yes, the kinetic energies can be equal. Since the kinetic energy is
proportional to mass and proportional to the square of the speed
1
2
(K mv ), if the car has twice the speed of the truck (a factor
2
of 4 contribution to the kinetic energy), then the kinetic energies
can be equal if the truck has 4 times the mass of the car.