A disk with a radial line painted on it is mounted on an axle perpendicular to i

Question

A disk with a radial line painted on it is mounted on an axle perpendicular to it and running through its center. It is initially at rest, with the line at theta_0 = -90 degree. The disk then undergoes constant angular acceleration. After accelerating for 3.1 s, the reference line has been moved part way around the circle (in a counterclockwise direction) to theta_t = 130 degree. Given this information, what is the angular speed of the disk after it has traveled one complete revolution (when it returns to its original position at -90 degree)? |omega| = rad/s

Explanation / Answer

given

Disc with radial line painted on it is initially at theta_0 = -90 degrees position

disc moving from rest,under constant angular acceleration,

after a time duration t = 3.1 s which is at the position theta_f = 130 degrees

finally the radial line reached its initial position (one complete revolution).

we know that the angular displacement in one completer revolution is zero radians

using the equations of motion in angular motion with constant angular acceleration ,

first

theta = w0*t+ 0.5*alpha*t^2

given W0=0 rad/s , theta = theta_f- theta _0 = (90+130)degrees = 210 degrees = 3.66519 rad

3.66519 rad = 0*t+ 0.5*alpha*3.1^2

alpha = 0.762787 rad/s2

when the line reaches the position 130 degrees its radial speed is

W_f^2 - W_0^2 = 2*alpha*theta

W_f^2 = 0+2*0.762787*3.66519

W_f = sqrt(2*0.762787*3.66519) rad/s

W_f = 2.364639rad/s

this is the speed when the line reaches the position 130 degrees , from then the line reaches the original potions -90 degrees , treating it as initial speed, so that it can cover 140 degrees displacement to reach -90 degrees position

after one complete revoultion,

so W_F^2 - W_f^2 = 2* alpha*theta

W_F^2 = 2.364639^2 + 2* 0.762787*2.44346

W_F = 3.053 rad/s

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