MAE 101 homework 4 sol

# MAE 101 homework 4 sol - MAE 101 Homework 4 Solution...

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MAE 101: Homework 4 Solution October 11, 2011 Problem 10. (Problem 3.11 from Gross et al). A uniform pole (length l , weight G ) leans against a corner as shown in the ﬁgure. A rope S prevents the pole from sliding. All surfaces are smooth. Determine the force S in the rope. Solution: The FBD is shown in the following ﬁgure. Because friction is negligible, all the contact forces will be perpendicular to the contact plane. Note that the contact force N 1 and N 2 must intersect at point O . The distance OA can be calculated as OA = h/ sin α/ sin α = h sin 2 α 1

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Take the moment about point O we have X M O = S × h sin 2 α - G × l 2 cos α = 0 Solving for S , we have S = Gl 2 h sin 2 α cos α . Problem 11. (Problem 3.13 from Gross et al). Two smooth rollers (each having weight G and radius r ) are connected by a rope (length a ) as shown in the ﬁgure. A lever (length l ) subjected to a vertical force F exerts contact forces on the rollers. Determine the contact forces between the rollers and the horizontal plane. Answer N 1 = G - F l a r 1 - 4( r a ) 2 , N 2 = G + F l a r 1 - 4( r a ) 2 where N 1 and N 2 are the contact forces acting on the left and on the right roller, respectively. Solution: Assume the angle between the lever and the cable to be α , and from geometry we have sin α = a 2 r and cos α = q 1 - ( 2 a r ) 2 . Let’s ﬁrst analyze the lever. The FBD of the lever is drawn as below. The lever has contact forces from the rollers which are perpendicular to it and also contact force F 0 from the ground and a load F , as shown. Consider the force equilibrium along the lever we must have F sin α = F 0 sin α , so that we have F 0 = F . Let’s know consider the FBD of the whole system. Assume contact force between roller A and ground is N 1 and roller B is N 2 . Draw the FBD of the whole system Let’s calculate the moment about point A . The original problem’s plot is not very clear where the left end of the lever lies, however we don’t have to know that if we consider the two F ’s from the ground and the load as a couple. To calculate the moment generated by those two
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MAE 101 homework 4 sol - MAE 101 Homework 4 Solution...

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