(e) (15 pt) Consider that P is moving with a constant speed of v along ey. Compu

Question

(e) (15 pt) Consider that P is moving with a constant speed of v along ey. Comput , using U, , and h. (30 pt in total) Consider the following system. At t = 0, a cart, C. starts to mov with a constant speed of vo on a horizontal track supported by a ceiling. The cart ha a pendulum system in which a particle, P, is supported by a string with a length o L. Two reference frames are defined. The inertial frame I is defined using e, and ey which are always fixed at O. At a given time t, we observed that the angle of ex an the vertical direction was , and the angle velocity was

Explanation / Answer

a. From the diagram

position of P at time t using ex and ez, given constant velocity vo

P = (vo*t + Lsin(theta))ex + L ez

b. let coordinates in frame R be (x,y)

then coordinates in frame I are (X,Y)

the relation can be defined as

X = xcos(theta) + ysin(thjeta)

Y = xsin(theta) - ycos(theta)

so

[cos(theta) sin(theta)][x] = [X]

[sin(theta) -cos(theta)][y] [Y]

so the transformation matrix is

[cos(theta) sin(theta)]

[sin(theta) -cos(theta)]

c. velocity of P at time t using ex and ey

v = (v + wLcos(theta))ex + wLsin(theta)ey

where w = d(theta)/dt

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