A particle of mass m rests on a horizontal 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 to be applied.

Explanation

Let a force F be applied at `angle theta` to move the body , as shown in Here , f represents the force of friction
For horizontal equilibrium ,
`F sin theta + R , = mg - F sin theta … `
put in (i) `F cos theta = mu (mg - F sin theta )`
`F (cos theta + mu sin theta ) = mu mg `
`F = F = (mu mg )/(cos theta + mu sin theta )`
Now , F will be minimum , when denominator ,
`(cos theta + mu sin theta )` = maximum
` :. (d)/(d theta) (cos theta + mu cos theta = 0 `
` - sin theta + mu cos theta = 0 `
or `tan theta = mu`
`sin theta (mu)/(sqrt(mu^(2) + 1 )), cos theta (1)/(sqrt(mu^(2) + 1`
put in (iii) , ` F_(min) = (mu mg)/(1/(sqrt(mu^(2)+1))+ (mu^(2))/(sqrtmu^(2) + 1 ))=(mumg)/(sqrt(mu^(2)+ 1`
`theta = tan^(-1) (mu)`
This is the direction of F min

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