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Download Applied Mechanics - Dynamics by Housner G & Hudson D PDF

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By Housner G & Hudson D

1950 - utilized Mechanics Dynamics, through George W. Housner & Donald E. Hudson (Division of Engineering, California Institute of Technology), D. Van Nostrand Co., Inc., Toronto, N.Y. & London

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Solution. A polar coordinate system will be convenient for the solution of this problem. 1. Given a particle of mass m moving in a circular path of radius 7 with a velocity of constant magnitude v . Find the force required to maintain this motion. Do this problem in two ways: (a) by describing the motions in a rectangular coordinate system whose origin is at the center of the circular path and (b) by using a plane polar coordinate system to describe the motion. Compare the two procedures. 2. A block W Aweighing 15 lbs and a block W B weighing 25 lbs slid(: down a 30" inclined plane as shown in thc figure.

The displacement of a point P (Fig. 1) is described by the magnitude and direction of the radius vector r which extends from the origin of a coordinate system to the point P. At time t let the displacement be r then at time t + At the displacement is r + Ar where Ar is the vector from P to P'. Between P and P' the average change of r per unit time is Ar/At and the velocity at P is obtained by taking the limiting value of Ar/At as At approaches zero: The direction of v is tangent to the path of motion at P.

EXAMPLE 3. A simplified picture of the mechanism of a helicopter blade is shown in Fig. 17. The blade oscillates about the horizontal axis P-P',which is carried on a rotating disk OB so that the whole assembly rotates with a constant angular velocity w. The blade has a mean position defined by the line BC,which makes an angle 0 with the horizontal. At any time t the blade makes an angle with the line BC, where is given by the equation = 40sin pt, # being the angular frequency of the "flapping" oscillation of the + + + blade.

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