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 part AB is smooth, speed of block of `B=3(m)//(s)`
K.E. Of block at `B=(1)/(2)xx0.5xx(3)^2=2.25J`
Work done against friction on part BD is
`mu_kmgxxBD=0.20xx0.5xx10xx2.14=2.14J`
K.E. Of block at `D=2.25-2.14=0.11J`
Between D and E
`0.11=(1)/(2)kx^2+mu_kmgx`
`=(1)/(2)xx2xx x^2+0.2xx0.5xx10x`
`=x^2+x`
`x^2+x-0.11=0`
`x=(-1+-sqrt(1+0.44))/(2)impliesx=0.1m`
when the spring is compressed by 0.1m the force exerted by spring is
`F_S=kx=2xx0.1=0.2N`
`(f_s)_(max)=mu_Smg=0.22xx0.5xx10=1.1N`
As `F_Slt(f_s)_(max)`, the block will not move.
Total distance traveled by the block is
`AB+BD+x=2+2.14+0.1`
`=4.24m`