Question

A 1 kg body slides from the point A along a horizontal track with an initial speed of $6m{s}^{-1}$ towards a weightless horizontal spring of length 2 m and force constant $4N{m}^{-1}$.The part AB of the track is frictionless and the part BC has the coefficients of static and dynamic friction 0.22 and 0.2, respectively. Find the total distance through which the block moves before it comes to rest completely ($g=10m{s}^{-2}$),
if AB = 4 m and BD = 4.28 m.

Answer 1

Solution Let total distance travelled be

(4+x)m.

As spring force is conservation

and friction is non-conservative.

$△KE+△PE$ = w friction.

w friction = - Dynamic fraction $\times$ distance

in friction through surface

w friction = $-\left(0.2\right)\left(1\right)\left(10\right)(\times )$

w friction = -2x

Initially,

$KEi=\frac{1}{2}m{v}_1^2=\frac{1}{2}\times 1\times (6{)}^2=18J$

$PEi=\frac{1}{2}k(0{)}^2=0$

finally,

$K{E}_f=\frac{1}{2}m{v}_{f}2=\frac{1}{2}\times 1\times (0{)}^{2}20J$

$P{E}_f=\frac{1}{2}k(0{)}^{2}20$

So, $(0-18)+(0-0)=-2x$

$x=9m$

Distance travelled = $4+9=13m$