A prismatic block of mass `m` is kept on a groove . The bottom line of the groove makes an angle `theta` with horizontal . The angle between is `phi` If the groove is symmetrical with the normal to the inclined plane containing the bottom line of the groove

a. Find the coefficient of friction `mu_(0)` between the block and groove so that the block begins to slide.
b. If `mu gt mu_(0)` find the friction force on the block.
c. If `mu lt mu_(0)` find the acceleration of the block.

Answer

`(tan theta)/(2) sin ((phi)/(2))`, b. `mg sin theta` c. ` g sin theta[1 - (mu)/(mu_(0))]`

Explanation

a. `N sin ((phi)/(2))= mg cos theta`

`mg sin theta= 2 mu_(0)N = (2 mu_(0) cos theta)/(sin (phi//2))`
`rArr mu_(0) = (tan theta)/(2) sin ((phi)/(2))`
b. if `mu gt mu_(0)`, than block will not slide and friction will be `f_(s) = mg sin theta`
c. if `mu lt mu_(0)`
`mg sin theta - 2 mu N = ma`
`rArr a = g sin theta -(2 mu)/(m) (mg sin theta)/( 2mu_(0)) = g sin theta[1 - (mu)/(mu_(0))]`