In the figure, point P is on the rim of a wheel of radius 2.0 m. At time t = 0,

Question

In the figure, point P is on the rim of a wheel of radius 2.0 m. At time t = 0, the wheel is at rest, and P is on the x-axis. The wheel undergoes a uniform counterclockwise angular acceleration of 0.010 rad/s2 about the center O.

(a) At time t = 0, what is the tangential acceleration of P?

(b) What is the linear speed of P when it reaches the y-axis?

(c) What is the magnitude of the net linear acceleration of P when it reaches the y-axis?

(d) How long after starting does it take for P to return to its original position on the x-axis?

This question was previously answered, but without showing the work. I don't understand the algebraic relationship between the tangential speed, linear speed, magnitude, and time. In other words, once you know tangential speed, how do you work your way to find the other unknowns?

Explanation / Answer

a) the tangential acceleration of P:

a = angular acceleration*r = 0.01*2= 0.02 m/s2

b) The angular displacement when it reaches y,

angular displacement = pi/2

angular speed when it reaches y is [w2 = w02 + 2*alpha*angular displacement],

Initial angular speed = w0 = 0

w = sqrt(2*0.01*pie/2)= 0.177 rad/s

so linear speed= wr = 0.177*2 = 0.354 m/s

c)

linear acce= sqrt(tangential^2+ centropetal^2)= sqrt(0.02^2 + 0.177^2/2)= 0.126 m/s2

d) to return back at x axis, angle= 2*pi

Hence, the required time is,

t = sqrt(2*2*pi/alpha(tangential acceleration)) = 35.44 s

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