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334 CHAPTER 10 Systems of Particles

*49. Mount Fuji has approximately the shape of a cone. The half-
angle at the apex of this cone is 65 , and the height of the apex
is 3800 m. At what height is the center of mass? Assume that
the material in Mount Fuji has uniform density.
*50. Show that the center of mass of a uniform flat triangular plate
is at the point of intersection of the lines drawn from the ver-
tices to the midpoints of the opposite sides.
*51. Consider a man of mass 80 kg and height 1.70 m with the
mass distribution described in Fig. 10.17. How much work
does this man do to raise his arms from a hanging position to
a horizontal position? To a vertically raised position?
*52. Suppose that a man of mass 75 kg and height 1.75 m runs in
place, raising his legs high, as in Fig. 10.35. If he runs at the
rate of 80 steps per minutes for each leg (160 total per
FIGURE 10.36 A hemispherical shell used as a gong.
minute), what power does he expend in raising his legs?

From this, deduce that the time-average height of a projectile
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released from the ground and returning to the ground is of
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its maximum height. (This theorem is useful in the calculation
0.521L of the average air pressure and air resistance encountered by a
0.418L projectile.)
0.254L
10.3 The Motion of the Center of Mass
57. A proton of kinetic energy 1.6 10 13 J is moving toward a
proton at rest. What is the velocity of the center of mass of the
FIGURE 10.35 Man with raised leg.
system?
58. In a molecule, the atoms usually execute a rapid vibrational
*53. A lock on the Champlain Canal is 73 m long and 9.2 m wide; motion about their equilibrium position. Suppose that in an
the lock has a lift of 3.7 m—that is, the difference between the isolated potassium bromide (KBr) molecule the speed of the
water levels of the canal on one side of the lock and on the potassium atom is 5.0 10 m/s at one instant (relative to the
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other side is 3.7 m. How much gravitational potential energy center of mass). What is the speed of the bromine atom at the
is wasted each time the lock goes through one cycle (involving same instant?
the filling of the lock with water from the high level and then
59. A fisherman in a boat catches a great white shark with a har-
the spilling of this water to the low level)?
poon. The shark struggles for a while and then becomes limp
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*54. The Great Pyramid at Giza has a mass of 6.6 10 metric when at a distance of 300 m from the boat. The fisherman
tons and a height of 147 m (see Example 7). Assume that the pulls the shark by the rope attached to the harpoon. During
mass is uniformly distributed over the volume of the pyramid. this operation, the boat (initially at rest) moves 45 m in the
(a) How much work must the ancient Egyptian laborers have direction of the shark. The mass of the boat is 5400 kg. What
done against gravity to pile up the stones in the pyramid? is the mass of the shark? Pretend that the water exerts no
(b) If each laborer delivered work at an average rate of 4.0 friction.
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10 J/h, how many person-hours of work have been stored 60. A 75-kg man climbs the stairs from the ground floor to the
in this pyramid? fourth floor of a building, a height of 15 m. How far does the
**55. A thin hemispherical shell of uniform thickness is suspended Earth recoil in the opposite direction as the man climbs?
from a point above its center of mass as shown in Fig. 10.36. 61. A 6000-kg truck stands on the deck of an 80000-kg ferry-
Where is that center of mass? boat. Initially the ferry is at rest and the truck is located at its
**56. Suppose that water drops are released from a point at the edge front end. If the truck now drives 15 m along the deck toward
of a roof with a constant time interval t between one water the rear of the ferry, how far will the ferry move forward rela-
drop and the next. The drops fall a distance l to the ground. If tive to the water? Pretend that the water has no effect on the
t is very short (so the number of drops falling though the air motion.
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at any given instant is very large), show that the center of mass 62. While moving horizontally at 5.0 10 m/s at an altitude of
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of the falling drops is at a height of l above the ground. 2.5 10 m, a ballistic missile explodes and breaks apart into
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