Question

An object is kept at x = 0, force on the object is F(const.) from t = 0. friction coefficient very as $\mu =cx$. Find the velocity after traveling half of total distance travelled.

Answer 1

Block cover $2y$ displacement at $2y$ displacement.

When frictional force($f$) equal to $F$

$f=\mu Mg=cyMg=F$

$y=\frac{F}{cMg}$

work done by $F$ is$W_1$;

$W_1$$=Fy=\frac{F^2}{cMg}$

Now, work done by the frictional force is $W_2$

$dw=fdx=\mu Mgdx$

$dw=cMgxdx$

$W_2={\int }_0^{y}dw=cMg{\int }_0^{\left(\frac{F}{cMg}\right)}xdx$

$W_2=cMg[\frac{x^2}{2}{]}_0^{\left(\frac{F}{cMg}\right)}=cMg\times \frac{F^2}{2\times (cMg{)}^2}$

$W_2$$=\frac{F^2}{2cMg}$

Now K.E of block B is$\frac{1}{2}M{v}^2$

By energy conservation

$W_1=W_2+\frac{1}{2}M{v}^2$

$\frac{F^2}{cMg}$$=\frac{F^2}{2cMg}$$+\frac{1}{2}M{v}^2$

$\frac{1}{2}M{v}^2$$=\frac{F^2}{2cMg}$

$v=\frac{F}{M}\sqrt{\frac{1}{cg}}$