Calculate the minimum force required to drag a body of mass m, resting on a horizontal surface. The coefficient of friction between the body and the surface is `mu`.

Explanation

Let the applied force be F, acting at an angle `theta` with the horizontal Fig.
`f_(2)` = force of limiting friction, mg = weight of the body and R = normal force.
For equilibrium,
`R+F sintheta = mg " ""or"`, R = mg - F sin`theta`
and F cos `theta = f_(s) = mu R= mu (mg- F sin theta`)
or, F cos `theta +muF sin theta = mu`mg
`:. F = (mu mg)/(cos theta+musin theta)`
Value of F will be minimum when ( cos`theta +mu sin theta`) is maximum.
`:. (d)/(d theta) (cos theta +mu sin theta) = 0 `
or , - sin `theta +mucos theta = 0 " ""or", tan theta = mu `
`:. sin theta = (mu)/(sqrt(1+mu^(2))),"and cos"theta = (1)/(sqrt(1+mu^(2)))`
`:. F_("min")= (mumg)/((1)/(sqrt(1+mu^(2)))+(mu^(2))/(sqrt(1+mu^(2))))=(mumg)/(sqrt(1+mu^(2)))" ""and"theta= tan^(-1)mu`