Question

A block of mass $m$ moving with a velocity $v_0$ on a rough horizontal surface from $x=0$. Coefficient of friction is given by $\mu ={\mu }_0+ax$. Find the kinetic energy of the block as a function of $x$ before it comes to rest.

Answer 1

Frictional force $=\mu N=\mu mg=({\mu }_0+\alpha x)mg$
$W=$ work done by the frictional force
$=-{\int }_0^x({\mu }_0+\alpha x)mgdx=-mg({\mu }_0.x+\frac{\alpha x^2}{2})$
Applying work energy theorem
Work done $=\Delta K$
$-mg({\mu }_0.x+\frac{\alpha x^2}{2})=\frac{1}{2}m{v}^2-\left(\frac{1}{2}\right)m{v}_0^2$
$\frac{1}{2}m{v}^2=\left(\frac{1}{2}\right)m{v}_0^2-mg({\mu }_0.x+\frac{\alpha x^2}{2})$