A solid marble of mass m = 25 kg and radius r = 6 cm will roll without slipping

Question

A solid marble of mass m = 25 kg and radius r = 6cm will roll without slipping along the loop-the-looptrack shown in the figure if it is released from rest somewhere onthe straight section of track. From what minimum height h above thebottom of the track must the marble be released to ensure that itdoes not leave the track at the top of the loop? The radius of theloop-the-loop is R = 1.00 m. If the marble is released from height 6R abovethe bottom of the track, what is the horizontal component of theforce acting on it at point Q?
A solid marble of mass m = 25 kg and radius r = 6cm will roll without slipping along the loop-the-looptrack shown in the figure if it is released from rest somewhere onthe straight section of track. From what minimum height h above thebottom of the track must the marble be released to ensure that itdoes not leave the track at the top of the loop? The radius of theloop-the-loop is R = 1.00 m. If the marble is released from height 6R abovethe bottom of the track, what is the horizontal component of theforce acting on it at point Q?

Explanation / Answer

This is an energy and circular motion problem. .    Initial potential energy = finalKE + final PE .       m g h    =    (1/2) m v2   +  (1/2) I 2   +   m g2R .       m gh      = (1/2) mv2    + (1/2) (2/5) m r2(v/r)2     + 2 m g R .        gh     =    0.7v2    + 2 g R .    Also...   using circular motion at thetop:          n + mg   = mv2 / R .       Since minimum n iszero,      v2 = gR    and .       g h = 0.7 gR   + 2 gR             h = 2.7 R   = 2.7 * 1.00 =   2.70 m   is theminimum starting height . For part b, you did not provide the picture. I will assumethat "point Q" is halfway up the loop. In that case... .        initial PE = final KE + final PE .        m g 6R   = 0.7 m v2   +   m g R .         5gR   = 0.7 v2 . Normal force provides centripetal force, so .        n = mv2 / R   =    m (5gR/0.7) /R    =  25* 5 * 9.80 / 0.7 =    1750N . Are you SURE the mass is    25 kilograms???      There is no knownmaterial that is dense enough to create a 25 kg sphere of only 6 cmradius. Please check the details of the problem. If my answers areincorrect, it is because you typo-ed the information.

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