Suppose a 0.250 kg ball is thrown at 10.0 m/s to a motionless person standing on

Question

Suppose a 0.250 kg ball is thrown at 10.0 m/s to a motionless person standing on ice who catches it with an outstretched arm as shown in Figure 9.31.

(a) Calculate the final linear velocity of the person, given his mass is 80.0 kg.
.0312 m/s
(b) What is his angular velocity if each arm has a 5.00 kg mass? You may treat his arms as uniform rods of length 0.9 m and the rest of his body as a uniform cylinder of radius 0.190 m. Neglect the effect of the ball on his rotational inertia and on his center of mass, so that it remains in his geometrical center.
  _______ rad/s ???
(c) Compare the initial and final total kinetic energy.
_______ (initial energy / final energy) ???

Explanation / Answer

m = mass of person = 80 kg

M = mass of ball = 0.250 kg

Vib = initial speed of ball = 10 m/s

Vip = initial speed of person = 0 m/s

V = final speed of person + ball

a)

Using conservation of momentum

m Vip + M Vib = (m + M) V

80 x 0 + 0.250 x 10 = (80 + 0.250) V

V = 0.0312 m/s

b)

mass of cylinderical body = Mc = 80 - mass of two arms = 80 - 10 = 70 kg

radius of cylindrical body = Rc = 0.19 m

moment of inertia of cylinder = Mc Rc2/2 = 70 (0.19)2 /2 = 1.2635 kgm2

mass of both arm = Ma = 10 kg

length of both arm = L = 2 x 0.9 = 1.8 m

moment of inertia of cylinder = Ma L2/12 = 10 (1.8)2 /12 = 2.7 kgm2

Combined moment of inertia of whole body = I = 2.7 + 1.2635 = 3.9635 kgm2

moment of inertia of ball after catching = Ib = M (L/2)2 = 0.250 (0.9)2 = 0.2025 kgm2

Using conservation of angular momentum

M Vib (L/2) = IW + IbW

(0.250) (10) (0.9) = (3.9635 + 0.2025) W

W = 0.54 rad/s

c)

initial total kinetic energy is only due to the ball which is given as

KE of ball = KEi = (0.5) M Vib2 = (0.5) (0.250) (10) = 1.25 J

final kinetic energy is the combination of translational and rotational kinetic energy , which is given as

KEf = (0.5) (M + m) V2 + (0.5)(I + Ib) W2

KEf = (0.5) (80 + 0.250) (0.0312)2 + (0.5) (3.9635 + 0.2025) (0.54)2

KEf = 0.6465 J

Ratio = KEi / KEf = 1.25 / 0.6465 = 1.93

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