A block of mass `m` rests on a rough floor . The coefficient of friction between the block and the floor is `mu`
a. Two boys apply force `P` at an angle `theta` to the horizontal. One of them pushes the block , the other one pulls. Which one would reqire less efferts to cause impending motion of the block?
b. What is the minimum force required to move the block by pulling it ?
c. Shown that the block is pushes at a certain angle `theta _(0)` it cannot be moved fro whatever the value of `P` be

Explanation

a.
In first case `N_(1) = mg - p sin theta`
In second case `N_(2) = mg + p sin theta`
Since normal force in second case is greater so the friction will be greater in second case when the block impends the motion Hence in first case , less effort is required
b.` P cos theta = mu N_(1) = mu (mg - P sin theta) rarr` for first case
`rArr P = (mu mg)/(cos theta + mu sin theta ) rArr p_(min) = (mu mg)/(sqrt(1 + mu^(2))`
c. For pushing . ` p cos theta gt mu N_(2)`
` rArr p cos theta gt mu(mg + p sin theta)`
`rArr p gt (mu mg)/(cos theta - mu sin theta)`
As `theta` increased `cos theta mu sin theta`decreases and hence `P` increase. Let as `theta = theta _(0)` denominator because zero so
`cos theta _(0)-mu - sin theta _(0) = 0 rArr theta _(0) = cot^(-1) (mu)`
At `theta = theta _(0) ,P gt oo ` which is not possible .