Question

The coefficient of friction between the block and horizontal surface is $\mu$. The block moves towards right under action of horizontal force $F$ (figure-a). Sometime later another force $P$ is applied to the block making an angle $\theta$ (such that $\tan \theta =\mu$) with vertical as shown in (figure-b). After application of force $P$, the acceleration of block should increase, decrease or remain same? Justify your answer.

Answer 1

$f_1=$ frictional force

$f_1=\mu mg$

acceleration $a_1$ is given below;

$F-f_1=m{a}_1$

$a_1=\frac{F-\mu mg}{m}$

When $F$ is applied then,

In this case, frictional force$f_2$;

$f_2=\mu mg+P\cos \theta$

and let acceleration is $a_2$

$F+P\sin \theta -f_2=m{a}_2$

$F+P\sin \theta -(\mu mg+P\cos \theta )=m{a}_2$

$a_2=(F-\mu mg)+\frac{(P\sin \theta -\mu P\cos \theta )}{m}$

$a_2=a_1+\frac{(P\sin \theta -\mu P\cos \theta )}{m}$

It is clear that

1) If $\sin \theta >\mu \cos \theta$, then $a_2>a_1$ (increase)

2) If $\sin \theta =\mu \cos \theta$, then $a_2=a_1$ (equal)

3) If $\sin \theta <\mu \cos \theta$, then $a_2<a_1$ (decrease)