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Unformatted text preview: Chapter 4 Kinetics of a System of Particles Question 41 A particle of mass m is connected to a block of mass M via a rigid massless rod of length l as shown in Fig. P4-1. The rod is free to pivot about a hinge attached to the block at point O . Furthermore, the block rolls without friction along a horizontal surface. Knowing that a horizontal force F is applied to the block and that gravity acts downward, determine a system of two differential equations describing the motion of the block and the particle. F g l m x M O P Figure P4-1 130 Chapter 4. Kinetics of a System of Particles Solution to Question 41 Kinematics Let F be a reference frame fixed to the block. Then, choose the following coordinate system fixed in reference frame F : Origin at O at t = E x = To the Right E z = Into Page E y = E z E x Next, let A be a reference frame fixed to the rod. Then, choose the following coordinate system fixed in reference frame A : Origin at O e r = Along OP e z = Into Page e = E z e r We note that the relationship between the basis { E x , E y , E z } and { e r , e , e z } is given as E x = sin e r + cos e (4.1) E y = - cos e r + sin e (4.2) Also, we have that e r = sin E x- cos E y (4.3) e = cos E x + sin E y (4.4) Using the bases { E x , E y , E z } and { e r , e , e z } , the position of the block is given as r O = x E x (4.5) Then the velocity and acceleration of the block in reference frame F are given, respec- tively, as F v O = x E x (4.6) F v O = x E x (4.7) Next, the position of the particle is given as r = r P = r O + r P/O = x E x + l e r (4.8) Next, the angular velocity of reference frame A in reference frame F is given as F A = e z (4.9) 131 The velocity of point P in reference frame F is then given as F v P = F d dt ( r O ) + F d dt ( r P/O ) = F v O + F v P/O (4.10) Now we already have F v O from Eq. (4.6). Next, since r P/O is expressed in the basis { e r , e , e...
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This note was uploaded on 04/03/2012 for the course AERO 310 taught by Professor Chakravorty during the Spring '07 term at Texas A&M.

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ilovepdf.com_split_6 - Chapter 4 Kinetics of a System of...

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