A `0.5kg` block slides from the point A on a horizontal track with an initial speed `3 m//s` towards a weightless horizontal spring of length `1m` and force constant `2N//m.` The part AB of the track is frictionless and the part BC has the coefficient of static and kinetic friction as '0.22' and `0.20` respectively. If the distances AB and BD are `2m` and `2.14m` respectively, find total distance through which the block moves before it comes to rest completely. `(g=10 m//s^(2) ).

Explanation

As the track AB is frictionless, the block moves this distance without loss in its initial
KE = `(1)/(2) mv^(2) = (1)/(2) xx 0.5 xx 3^(2) = 2.25 ` J, In the path
BD as friction is present, so work doen against friction
`= mu_(k) "mgs" = 0.2 xx 0.5 xx 10 xx 2.14 = 2.14 `J
so, at D the KE of the block is = 2.25 0 2.14 = 0.11 J.
now, if the spring is compressed by x
0.11 = `(1)/(2) xx k xx x^(2) + mu_(k)` mgx
i.e., ` 0.11 = (1)/(2) xx 2 xx x^(2) 0.2 xx 0.5 xx 10 x `
or `x^(2) x - 0.11 ` = 0
which on solving gives positive value of x = 0.1 m After moving the distance x = 0.1 m the block comes to rest. Now the compressed spring exerts a force
F = kx = `2 xx 0.1 = 0.2` N
On the block while limiting frictional force between block and track is
`f_(L) - mu_(s)` mg = `0.22 xx 0.5 xx 10 = 1.1` N.
Since, F `lt f_(L)`. The block will not move back. so, the total distance moved by block
= AB + BD + 0.1
= 2 + 2.14 + 0.1 = 4.24 m