A `0.5` kg block sides from the point `A` (see fig) on a hurizontal track as wil initial speed of `3` m//s toward a weightes horizental spring of length `1` in and force constant `2` newtav in the part `AB` of the track is frictionless friction and the per `BC` has the coefficient of the state `AB` and and `BD`are `2` m and `2.14` m repectively , find the total distance through which the block move before if conseto rest completely (Taking` = 10m//s^(2)) ` .

Answer

B, D

Explanation

K.E of black = work against friction ` + P.E. spring
`(1)/(2) mv^(2) = mu mg (2.14 + x ) b+ (1)/(2) kx^(2)`
` (1)/(2) xx 0.5 xx 3^(2) = 0.5 xx 9.8 (2.14 + x) + (1)/(2) 2 xx x^(2) `
`2.14 + x + x^(2) = 2.25`
` :. X^(2) + x - 0.11 = 0`
On solving we get `x = - (11)/(10) ` or `x = (1)/(10) = 0.1` (valid answer )
Here the body stop moment
Reatoring force at `Y = kx = 2 xx 0.1 = 0.2 N`
Frictional force at `Y = mu _(2) mg = 0.22 xx 0.5 xx 9.8 = 1.078 N `
Since friction gt restoring force , the body will step here . The total distance travelled
` = AB +BD + DY = 2 + 2.14 + 0.1 = 4.24 m`