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. The first ball has an angular speed of 21.2 rad/s and the second ball has an angular speed of 18.3 rad/s. The first ball, which has a radius of 5.88 cm, 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 4.48 cm and is rolling along the stationary floor. If the second ball starts out 5.69 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

We need to first find the total speed of both balls. We need to convert the angular speed to linear speed by multiplying by the radius. The velocity of the first ball will be: v1=?1r1+vbelt=20.2(.0613)+2.19 v1=3.42826 m/s Now to find the time: t1=d1/v1=8.87/3.4286 t1=2.587 seconds The velocity of the second ball will be v2=?2r2=17.3(.0468) v2=.80964 Now the time is: t2=d2/v2=6.74/.80964 t2=8.325 seconds In order for the second ball to find in the same time as the first ball it would have to travel at: v=d2/t1=6.74/2.587 v=2.605 m/s Converting that in angular speed gives; ?=v/r2=2.605/.0468 ?=55.67 rad/sec

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