A tennis ball is a hollow sphere with a thin wall. It is set rolling without sli

Question

A tennis ball is a hollow sphere with a thin wall. It is set rolling without slipping at 3.93 m/s on a horizontal section of a track as shown in the figure below. It rolls around the inside of a vertical circular loop of radius r = 48.3 cm. As the ball nears the bottom of the loop, the shape of the track deviates from a perfect circle so that the ball leaves the track at a point h = 21.0 cm below the horizontal section.

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A tennis ball is a hollow sphere with a thin wall. It is set rolling without slipping at 3.93 m/s on a horizontal section of a track as shown in the figure below. It rolls around the inside of a vertical circular loop of radius r = 48.3 cm. As the ball nears the bottom of the loop, the shape of the track deviates from a perfect circle so that the ball leaves the track at a point h = 21.0 cm below the horizontal section. Find the ball's speed at the top of the loop. Demonstrate that the ball will not fall from the track at the top of the loop. Find its speed as it leaves the track at the bottom. Suppose that static friction between ball and track were negligible so that the ball slid instead of rolling. Would its speed then be higher, lower, or the same at the top of the loop? Explain your answer.

Explanation / Answer

Applying principle of consrevation of momentum we ve

1/2 mv1^2 + 1/2 Iw1^2 = 1/2 mv2^2 + 1/2 Iw2^2 + mgy2

I = 2/3 mr^2 and w = v/r

substituting the values we ve

v2 = sqrt{v1^2 - 6/5*9.8 * y2}

v2 = sqrt{3.93^2 - 6/5 *9.8 *0.966}

V2 = 2.021 m/sec

b. Centripital accln = v2^2/r = 2.021^2/0.483 = 8.456 m/sec^2

hence its neccessary for ball to be in contact with track assuming track ppushing downward...

c. again using the same concept as that of a we ve

v3 = sqrt{v1^2 - 6/5*9.8 * y3}

v3 = sqrt{3.93^2 - 6/5 *9.8 *(-0.21)}

V3 = 4.232 m/sec

d. v2 = sqrt{v1^2 - 2gy2} = sqrt{3.93^2 - 2*9.8*0.966} =sqrt(-3.4887) -1.86 m/sec ...

neglecting the minus sign..

e. since v2 is imaginary we can say the ball starts with less energy compared to a and it never reaches the top...

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