Page 137 - Fisika Terapan for Engineers and Scientists

Answers to Checkups 337

Answers to Checkups

Checkup 10.1 3. No. The center of mass is a weighted average of position; such
an average can never be greater than all of the positions
1. We assume the usual case that the truck has a larger mass than
averaged.
the automobile. Then the equality of their momenta (mv)
4. (C) Shifts upward. If the heavy mass at the bottom falls off,
implies that the automobile has a larger speed (the ratio of the
the center of mass is higher. When the center of mass of the
velocities will be the inverse of the ratio of the masses). Since
2
1
1
the kinetic energy is mv mv v and the momenta sailboat is too high, it is top-heavy, and prone to tip over.
2
2
(mv) are equal, then the vehicle with the larger speed, the
automobile, will also have the larger kinetic energy. Checkup 10.3
2. They cannot have the same momentum, since the signs of
1. For the (isolated) system of person plus canoe, with no initial
their momenta will be opposite. They can have the same
motion, the center of mass stays at the same fixed position as
kinetic energy, since that depends only on the speed:
1
K mv 2 . you begin and continue your crawl. Thus as your mass moves
2
from the rear to the front, the boat moves backward a suffi-
3. No—the momentum, like the velocity, is also reversed, and it
cient distance to keep the center of mass of the combined
has the opposite sign after the impact.
system fixed.
4. Yes—for practical purposes, the Solar System is essentially an
2. No. In the extreme case where the boxcar has zero mass, you
isolated system, and so the net momentum is constant. The
remain fixed relative to the ground, and the boxcar rolls a dis-
net kinetic energy is not constant, since during motion, kinetic
tance equal to its length as you walk from the rear to the front.
energy is converted to potential energy and vice versa.
If the boxcar has appreciable mass, you will move toward your
2
2
1
1
5. (a) For any directions, the total kinetic energy is mv mv common center of mass, which will be a distance less than the
2
2
2
mv . For parallel motion (both southward), the total momen- length of the boxcar.
tum is mv mv 2mv southward. (b) The total kinetic energy
2 3. If the marbles were dropped vertically, then the center of mass
is again mv , while for antiparallel motion, the total momentum
remains fixed at the point of impact with the floor, even
is mv mv 0 (no direction). (c) The total kinetic energy is
2 though the marbles scatter in all directions.
again mv , while for perpendicular motion, the total momen-
2 2 4. (B) 5 m/s south. Using Eq. (10.37) with positive velocity
tum has magnitude 2(mv) (mv) 22mv and is
northward, v (M 25 m/s 2M 20m/s) / (3M )
directed 45 south of east. CM
( 15 m/s)/3 5 m/s.
6. (C) Automobile; truck.The truck has a larger mass M than the
automobile mass m. Let the truck speed be V and the automo-
2
2
1
1
bile speed be v.The equal kinetic energies ( MV mv ) Checkup 10.4
2
2
then imply that the automobile will have the larger speed
1/2
v (M/m) V. If we substitute one power of this v into the 1. No. The initial kinetic energy is large, and the final kinetic
1
kinetic energy equality and cancel a factor of V , we find energy is small or zero; the energy is transformed into other
2
1/2
MV (M/m) mv; thus, the truck momentum is larger. forms: elastic energy (deformation of automobile parts),
friction, sound, and heat.
Checkup 10.2 2. The total energy is conserved, if we consider only gravitational
potential energy and kinetic energy (in actuality, some other
1. Consider the average position of the mass distribution. For the energy is lost, for example, as the Sun’s light is radiated away
curved snake shown in the figure, the center of mass is at a into space). Neither kinetic nor potential energy is separately
point in the space below the top arc, perhaps slightly below conserved; these two are traded back and forth, for example, as
center (because of the two bottom arcs) and slightly to the the planets move in their elliptical orbits.
right of center (because of the head). 3. (A) P only. Since there is no net external force, the total
2. Consider the average position of the mass distribution. For the momentum of such an isolated system is always simply con-
horseshoe, the center of mass is below center in the space in served. However, the spring will stretch and compress during
the middle of the arc, along the vertical line of symmetry, at a the motion, trading kinetic for potential energy, so K and U
point well away from the open end. will not remain constant.