A rigid ring of R = 1.00 m is started in rotational counter-clockwise motion at

Question

A rigid ring of R = 1.00 m is started in rotational counter-clockwise motion at t = t_0 with constant angular acceleration alpha = 5.60 rad/s^2 about a fixed axis through its center. Calculate the magnitude and direction (relative to the x-axis shown) of a at point P at t = t_0, and draw a sketch of the vector on the figure below. Calculate the direction and magnitude of a at point P', which is the new location of P after it has been displaced pi rad. Sketch this vector on the figure below. Based on the results of this problem does constant alpha

Explanation / Answer

at point P

at time t = t0

initial angular speed wi = 0

tangential acceleration at = R*alpha = 1*5.6 = 5.6 m/s^2

radial acceleration ar = R*wi^2 = 0

magnitude of a = sqrt(at^2+ar^2) = 5.6 m/s^2

direction = vertically downwards ( along negative y axis)

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(b)

at point P'

final angular speed wf = sqrt(2*alpha*theta)

wf = sqrt(2*5.6*pi) = 5.93 rad/s

tangential acceleration at = R*alpha = 1*5.6 = 5.6 m/s^2

radial acceleration ar = R*wf^2 = 1*5.93^2 = 35.16 m/s^2

magnitude of a = sqrt(at^2+ar^2) = sqrt(5.6^2 + 35.16^2) = 35.6 m/s^2

direction , theta = tan^-1(at/ar)

direction , theta = 9.05 degrees

(c)

NO

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