Question

A block of mass $m$ initially has a velocity $v_0$ when it just touching a spring. The block moves through a distance $I$ before it stops after compressing the spring. The spring constant is $k$ and the coefficient of kinetic friction between block and table is $\mu$. As the block moves the distance $l$
(a) what is the work done on it by the spring force? Are there other forces acting on the block, and if so, what work do they do?
(b) what is the total work done on the block?
(c) use the work energy theorem to find the value of $l$ in terms of $v_0,\mu ,g$ and $k$

Answer 1

(a) The net force acting on the block by the spring is equal to $F_{sprinf}=kx$
Work done by the spring $\int {\overrightarrow{F}}_{spring}.d\overrightarrow{s}$
$=\int F_{sprinf}ds\cos {180}^o=-\int F_{sprinf}.ds=-\int kxdx=\frac{-k{l}^2}{2}$
(b) The total work done $W=\Delta KE=0-\left(\frac{1}{2}\right)m{v}_0^2=-\left(\frac{1}{2}\right)m{v}_0^2$
(c) The work done by friction $=-\mu mgl$
The total work done $W_s+W_f=\Delta KE\Rightarrow -\mu mgl-\left(1/2\right)k{l}^2\Rightarrow \frac{1}{2}k{l}^2+\mu mgl=\left(\frac{1}{2}\right)m{v}_0^2$
$\Rightarrow l=\frac{\mu mg}{k}[\sqrt{1+\frac{k}{m}{\left(\frac{v_0}{\mu g}\right)}^2}-1]$