A rod leans against a stationary cylindrical body as shown in figure, and its right end sides to the right on the floor with a constant speed `v`. Choose the correct option (s)

Options

A. the angular speed `omega` is `(_Rv^(2)(2x^(2)-R^(2)))/(x^(2)(x^(2)-R^(2))^(3//2))`
B. the angular acceleration `alpha` is `(Rv)/(xsqrt(x^(2)-R^(2)))`
C. the angular speed `omega` is `(Rv)/(xsqrt(x^(2)-R^(2)))`
D. the angular acceleration `alpha` is `(-Rv^(2)(2x^(2)-R^(2)))/(x^(2)(x^(2)-R^(2))^(3//2))`

Answer

C, D

Explanation

From the geometry, `x=R/(sin theta)`
Also, `omega=-(d theta)/(dt)`. Therefore,
`v=(dx)/(dt)=d/(dt)(R/(sin theta))`
`=(-R (d theta//dt)cos theta)/(sin^(2) theta)=(omegaR cos theta)/(sin^(2) theta)` ltbgt `omega=(v sin^(2) theta)/(R sin theta)=(Rv)/(xsqrt(x^(2)-R^(2)))`
`alpha=(d omega)/(dt)=d/(dt)((Rv)/(x(sqrt(x^(2)-R^(2))))=-(Rv^(2)(2x^(2)-R^(2)))/(x^(2)(x^(2)-R^(2))^(3//2))`