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322 CHAPTER 10 Systems of Particles

Suppose a man of average proportions performs a high jump,while
Concepts EXAMPLE 8
in arching his back (see the chapter opening photo). At the peak of
Context
his jump, his torso is approximately horizontal; his thighs, arms, and head make an
angle of 45 with the horizontal;and his lower legs are vertical,as shown in Fig.10.19b.
(a) How much is his center of mass shifted downward compared with a man who goes
over the pole horizontally (Fig.10.19a)? (b) How much is his potential energy reduced?
Assume the mass of the jumper is M 73 kg and his height L 1.75 m.
SOLUTION: (a) In Fig. 10.19a, the center of mass of the horizontal body is at
y 0, since each segment is essentially at y 0. In Fig. 10.20, we have used the
relative locations of the hinge points and centers of mass from Fig. 10.17 to deter-
mine the vertical position of each body segment in the arched-back position. For
example, the center of mass of the thigh is at a distance 0.521L 0.425L 0.096L
from the hip joint, and so is at a vertical distance 0.096L sin 45 0.068L below
y 0. Similarly, we can determine that the centers of mass of the lower legs, the
feet, the head, the upper arms, the forearms, and the hands are at y 0.270L,
0.434L, 0.016L, 0.067L, 0.183L, and 0.269L, respectively. From Fig.
10.17, the masses of all seven segments are 0.215M, 0.096M, 0.034M, 0.069M,
0.066M, 0.042M, and 0.017M, respectively. The torso, of mass 0.461M, is again
at y 0. Thus, using Eq. (10.19) or (10.22), the arched-back center of mass is at

1 n
y a m y
CM i i
M
i 1
1
(0.215 0.068 0.096 0.270 0.034 0.434 0.069
M
0.016 0.066 0.067 0.042 0.183 0.017 0.269
0 0.461)ML
0.073L 0.073 1.75 m 0.13 m

Thus a height advantage of 13 cm is gained in this arched position.

(b) According to Eq. (10.33), the potential energy is changed by

U Mg y
CM
2
73 kg 9.81 m s ( 0.13 m) (10.34)
93 J

(a) (b) Thighs, neck, and arms bend
Center of mass 45° at respective hinge points;
is at y = 0. lower legs are vertical.

y = 0 y = 0
45° 45°

Center of mass can
be calculated from
data of Fig. 10.17.

FIGURE 10.19 (a) Horizontal position. (b) High jumper in arched-back position.