A person is closing a heavy gate with rusty hinges by pushing the gate with car

Question

A person is closing a heavy gate with rusty hinges by pushing the gate with car A . If w = 23 m and nu A = 1.3 m/s, determine the angular velocity of the gate when 0 = 16degree. theta = rad/s. This problem is most easily approached by differentiating a geometrical constraint. Note that if we define y as the vertical distance between the closed gate configuration and the car at A . we have: y/w = tan Theta. Here omega = 23 m is a constant. Differentiating this expression once with respect to time and noting y = -vA provides an expression for the angular velocity of the gate. Theta.

Explanation / Answer

As given in the question as a hint

y = w tan => dy/dt = w sec2() d/dt = w sec2()

d/dt = . =

d/dt = dy/dt /(w sec2()) = -1.3 /(23*sec2()) = - 0.056 cos2()

when = 16o

d/dt = -0.0517 rad/s

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