Question

A block of mass m resists on a rough floor. The coefficient of friction between the floor and block 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 require less efforts to cause impending motion of the block?
b. What is the minimum force required to move the block by pulling it?

c. Show that if the block is pushed at a certain angle ${\theta }_0$, so that it cannot be move then what will be the value of P .

Answer 1

REF.Image.

$a)N=mg+psin\theta$

$b=\mu N$

$=\mu (mg+psin\theta )$

$b=\mu N$

$=\mu (mg+psin\theta )$

$F_{net}=Pcos\theta -\mu (mg+p\sin \theta )$

$N=mg-psin\theta$

$f=\mu N$

$=\mu (mg-Psin\theta )$

$F_{net}=Pcos\theta -\mu (mg-psin\theta )$

So here,

$F_{ne{t}_2}>F_{ne{t}_1}$

So, pulling is easiest than pulling.

b). For Req. min force in pulling

$F_{ne{t}_2}=0$

$=Pcos\theta -\mu (mg-psin\theta )$

$pcos\theta =\mu mg-\mu psin\theta$

$p(cos+\mu sin\theta )=\mu mg$

$P=\frac{\mu mg}{cos\theta +\mu sin\theta }$

(c) block is pushed at ${\theta }_0$

value of P

$F_{ne{t}_2}=0$

$0=Pcos\theta -\mu mg-\mu Psin$

$P=\frac{\mu mg}{\mu sin\theta -cos\theta }$