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 0.0713 m, is rolling along a conveyor belt which is moving at 2.19 m/s and starts out 8.57 m from the finish line. The second ball has a radius of 0.0448 m and is rolling along the stationary floor. If the second ball starts out 6.59 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

relative velocity of ball with respect to ground,

v1 = 2.19 + r1*w1

= 2.19 + 0.0713*24.2

= 3.915 m/s

time taken, t1 = d1/v1

= 8.57/3.915

= 2.19 s

velcoity of second ball with respect to griund, v2 = 0.0448*13.3

= 0.596 m/s
time taken for the seond ball, t2 = d2/v2

= 6.59/0.596

= 11.06 s

let w2 is the required angular speed,

t2 = d2/v2

2.19 = 6.59/(0.0448*w2)

==> w2 = 6.59/(0.0448*2.19)

= 67.2 rad/s

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