PHY183-Lecture35 - Loops ! ! A solid sphere is released...

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March 24, 2006 Physics for Scientists&Engineers 1 1 Physics for Scientists & Physics for Scientists & Engineers 1 Engineers 1 Spring Semester 2006 Lecture 35 March 24, 2006 Physics for Scientists&Engineers 1 2 Loops Loops ! A solid sphere is released from rest and rolls down an incline, from which it enters a circular segment of radius R . ! Question: What is the minimum height h from which the sphere has to be released, so that it does not fall off the track? ! Answer: At the top of the loop the normal force must be greater than zero to maintain contact with the track. The centripetal acceleration is larger than the gravitational acceleration. g < a c = v 2 R March 24, 2006 Physics for Scientists&Engineers 1 3 Recall: ! Analysis is exactly the same as for rotating mass on string. At top: If speed is too small, string goes slack before it reaches the top. Smallest speed corresponds to T=0. For rollercoaster, replace T with N. T mg T mg T + mg = mv 2 r March 24, 2006 Physics for Scientists&Engineers 1 4 Loops (2) ! The potential energy at h is ! The potential energy at the top of the loop is ! The kinetic energy is ! Conservation of energy gives us U 0 = mgh U = mg 2 R K = (1 + c ) 1 2 mv 2 c sphere = 2 5 K + U = + c ) 1 2 mv 2 + mg 2 R = K 0 + U 0 = mgh
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March 24, 2006 Physics for Scientists&Engineers 1 5 Loops (3) ! Solving for v 2 ! Putting in our condition that the ball not fall off the track ! Solving for h gives us ! Putting in c = 2/5 for a sphere v 2 = 2 g ( h ! 2 R ) / (1 + c ) R (1 + c ) < 2 h ! 4 R " h > 5 + c 2 R h > 5 + 2 5 2 R = 2.7 R g < v 2 R = 2 g ( h " 2 R ) R + c ) March 24, 2006 Physics for Scientists&Engineers 1 6 Math Insert: Vector Product ! Multiplication of vectors so far:
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This note was uploaded on 03/19/2008 for the course PHY 183 taught by Professor Wolf during the Spring '08 term at Michigan State University.

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PHY183-Lecture35 - Loops ! ! A solid sphere is released...

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