Two different balls are rolled (without slipping) toward a common finish line. T

Question

Two different balls are rolled (without slipping) toward a common finish line. Their angular speeds are shown to the right. The first ball, which has a radius of 0.0663 m, is rolling along a conveyor belt which is moving at 1.44 m/s and starts out 8.87 m from the finish line. The second ball has a radius of 0.0488 m and is rolling along the stationary floor. If the second ball starts out 6.74 m from the finish line, how long does each ball take to reach the finish line? What angular speed would the losing ball have needed to cross the finish line at the same time as the winning ball?

Explanation / Answer

here,

radius of ball 1, r1 = 0.0663 m
angular velocity of ball 1, w1 = 21.7 rad/s
distancec to finish line, d1 = 8.87 m

radius of ball 2, r2 = 0.0488
angular velocity of ball 2, w2 = 13.3 rad/s
distance to finish line, d2 = 6.74 m

veloicity of conveyer belt, v = 1.44 m/s

since direction of velocity is towards so :

total speed of ball 1, v1f = 1.44 * + 21.7 * 0.0663
total speed of ball 1, v1f = 2.878 m/s

time for ball 1, t1 = d1/v1f
time for ball 1, t1 = 8.87/2.878
time for ball 1, t1 = 3.082 s

total speed of ball 2, v2f = 1.44 * + 13.3 *0.0488
total speed of ball 2, v2f = 2.089 m/s

time for ball 2, t2 = d2/v2f
time for ball 2, t2 = 6.74/2.089
time for ball 2, t2 = 3.226 s

so ball 1 will need to have speed ,
v = 6.74 /(3.082)
v = 2.187 m/s

so
required angular velocity, w = v/r
required angular velocity, w = 2.187/0.0488
required angular velocity, w = 44.816 rad/s

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