Solution of Sheet (8) - 1 Solution of Sheet (8) The...

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1 Solution of Sheet (8) The Constrained Motion 1. A small particle having a mass m is placed inside a smooth circular vertical tube. If ) 3 ( min max υ = , determine the ratio between maximum and minimum normal force between the particle and tube, the velocity and the normal force when the particle reaches the end of horizontal diameter. Solution D External Forces B A C N C mg mg N A mg N B mgcos mg N mgsin B A C Effective Forces ma n ma t
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2 Given : v max = 3 v min …(1) , where v max = v A at = 0 o min = v C at = Applying the principal of work and energy between A & C ( ) ( ) ( ) ) 2 ....( 2 2 4 3 4 3 4 2 2 1 2 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 ga v ga v ga v v ga v ga v v a mg mv mv W mv mv C C C C C A C A C C A A C = = - = - - = - = - = - = - Substituting equation (2) in equation (1), we get: ) 3 ....( 6 2 3 ga ga v A = × = Applying Newton’s 2 nd law at point A, we get: ( ) ) 4 ....( 7 6 6 2 2 mg mg mg ga a m mg N a v m mg N ma F A A A n n = + = + = = - = Applying Newton’s 2 nd law at point C, we get: ( ) ) 5 ....( 2 2 2 2 mg mg mg ga a m mg N a v m mg N ma F C C C n n = + - = + - = = + = The ratio between N min max can be get by dividing equation (4) by equation (5), we get: ) 6 ....( 7 7 = = mg mg N N C A To get the normal reaction at the end of the horizontal diameter at point B, we apply the principal of work and energy between points A & B as follows: ( ) ) 7 ....( 2 4 2 6 2 2 1 2 1 2 1 2 1 2 2 2 2 2 2 2 ga v ga ga ga ga v v mga mv mv W mv mv B A B A B B A A B = = - = - = - = - = -
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3 Applying Newton’s 2 nd law at point B, we get: ( ) ) 8 ....( 4 4 2 2 mg ga a m N a v m N ma F B B B n n = = = = 2. A small particle having a mass m is placed inside a smooth circular vertical tube ( r=10cm ). If the particle starts its motion from the upper end of the vertical diameter with initial velocity sec) / mt 1 ( , determine velocity and the normal force at the lower end of the vertical diameter. Solution
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This note was uploaded on 12/21/2010 for the course ENGINEERIN mp108 taught by Professor Elbarki during the Spring '08 term at Alexandria University.

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Solution of Sheet (8) - 1 Solution of Sheet (8) The...

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