A blocky of mass m rests on a horizotal floor with which it has a coefficient of static friction `mu.` It is desired to make the body move by applying the minimum possible force F. Find the magnitude of F and the direction in which it has be applies.

Options

A. `mu mg `
B. `( sqrt(1+ mu^2))/(mu) mg`
C. `mu sqrt(1 + mu^2) mg `
D. `(mu m g )/( sqrt(1+mu^2))`

Answer

D

Explanation

Suppose the force F is applied at an angle `theta ` with horizontal, as shown in figure. Then

` R + F sin theta = mg or R = mg - F sin theta`
Force of friction `, f = mu R = mu ( mg - F sin theta)`
the block will move , when
` F cos theta ge f ` i.e., ` F cos theta ge mu mg - mu F sin theta `
` or F ( cos theta + mu sin theta) ge mu mg or F ge ( mu mg )/(cos theta + mu sin theta)`
Now ,F will be minimum when ` cos theta + mu sin theta = ` maximum for whcih
` (d)/( d theta)( cos theta + mu sin theta ) =0`
` or - sin theta + mu cos theta = 0 or tan theta = mu`
` therefore sin theta = ( mu )/( sqrt(1+ mu^2)) , cos theta = ( 1)/( sqrt(1+ mu^2))`
from (ii ),
` F ge (mu mg)/( (1)/( sqrt(1+mu^2)) +( mu^2)/(sqrt( 1+mu^2)))=(mu mg)/( sqrt(1 + mu ^2))`
` therefore F_(min) -=(mu mg)/( sqrt(1+ mu^2))`