13. Rotational Inertia & Energy

A solid sphere of mass M = 10 kg and radius R = 2 is rolling without slipping with speed V = 5 m/s on a flat surface when it reaches the bottom of an inclined plane that makes an angle of Θ = 37° with the horizontal. The plane has just enough friction to cause the sphere to roll without slipping while going up. What maximum height will the sphere attain? (Use g = 10 m/s².)

You may remember that the lowest speed that an object may have at the top of a loop-the-loop of radius R, so that it completes the loop without falling, is √gR . Determine the lowest speed that a solid sphere must have at the bottom of a loop-the-loop, so that it reaches the top with enough speed to complete the loop. Assume the sphere rolls without slipping.

(I) Calculate the translational speed of a uniform solid cylinder when it reaches the bottom of an incline 6.50 m high. Assume it starts from rest and rolls without slipping.

A marble of mass m and radius r rolls along the looped rough track of Fig. 10–77. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? (a) Assume r << R ; (b) do not make this assumption. Ignore frictional losses.

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