Find the Answer to the Ball's Velocity Puzzle

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The discussion centers on determining the necessary backward initial angular velocity (wo) for a ping pong ball to return to its original position with the same velocity (-Vo) after rolling without slipping. The conclusion states that wo must exceed 4Vo/R, where R is the radius and Vo is the initial speed. The time to return to the original spot is calculated as T = 2Vo/(g*µk), and the angular acceleration is given by α = (3/2)(g*µk)/R, derived from the moment of inertia for a hollow sphere.

PREREQUISITES
  • Understanding of rotational dynamics and angular velocity
  • Familiarity with the concepts of kinetic friction and its coefficient (µk)
  • Knowledge of the moment of inertia for a hollow sphere
  • Basic principles of motion under gravity (g)
NEXT STEPS
  • Study the moment of inertia calculations for different shapes, focusing on hollow spheres
  • Explore the relationship between linear and angular motion in rolling objects
  • Learn about the effects of friction on motion and rolling without slipping
  • Investigate the dynamics of projectile motion and its equations
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Physics students, mechanical engineers, and anyone interested in the dynamics of rolling motion and angular velocity calculations.

kamiltartar
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please help me!

I will be so happy if someone gives the answer to this question in a while.
A ping pong ball of radius R and mass M is started with an initial speed Vo and a backward initial angular velocity of wo.The coefficient of kinetic friction between the ball and the table is µk.What should wo be(in terms of R,M,Vo,µk) so that the ball comes back with the same velocity, i.e. the velocity -Vo when rolling without slipping starts?
 
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I get w_o>4 Vo/R.
 
The time to get back to the original spot is T=2Vo/g*mu.
The angular acceleration of the ball is alpha=(3/2)g*mu/R.
This comes from R*g*mu=I*alpha, using the I for a hollow sphere.
Thus w=w_o-(3/2)g*mu*t/R=w_o-3 Vo/R.
and w must be greater than Vo/R.
 

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