Question

A ball, which is initially at rest, starts rotating around a fixed point. The ball has a constant angular acceleration. You need to know that the radial acceleration of the ball aradarad is proportional to ball's angular displacement θθ.

Find the angular displacement of the ball if its resultant acceleration makes an angle of 35.9 ∘∘ with the radial direction?

Answer 1

given
Initial angular speed, wo = 0

let alfa is the angular acceleration.

use,

(w^2 - wo^2) = 2*alfa*theta

w^2 - 0^2 = 2*alfa*theta

w^2 = 2*alfa*theta

we know, tangential acceleration,
a_tan = r*alfa

we know, radial acceleration,
a_rad = r*w^2

tan(35.9) = a_tan/a_rad

tan(35.9) = r*alfa/(r*w^2)

= alfa/w^2

= alfa/(2*alfa*theta)

= 1/(2*theta)

==> theta = 1/(2*tan(35.9))

= 1/(2*tan(35.9))

= 0.691 radians (or) 39.6 degrees <<<<<<<------------Answer

Note : please comment for any further clarification.