Question

An elastic string of unit cross-sectional area and natural length `(a+b)` where `a gt b` and modulii of elasticity Y has a particle of mass m attached to it at a distance a from one end, which is fixed to a point A of a smooth horizontal plane. The other end of the string is fixed to a point B. so that string is just unstretched. If particle is diplacement towards right by. distance `x_(0)` and then released then

A. The time period of the oscillation will be `pi(sqrt(a)+sqrt(b))sqrt((m)/(Y))`
B. The time period of the oscillation will be `2pi(sqrt(a)+sqrt(b))sqrt((m)/(Y))`
C. The separation between two extreme positions will be `((sqrt(a)+sqrt(b))/(sqrt(a)))x_(0)`:
D. The separation between two extreme positions will be `((sqrt(a)+sqrt(b))/(sqrt(b)))x_(0)`:

Answer 1

Correct Answer - A::C

from `t_(B rarr 0)`
`T = y (x)/(a) rArr (d^(2)x)/(dt^(2)) = (-y)/(am) x rArr t_(B rarr 0) = (pi)/(2) sqrt((am)/(y))`
simply `t_(B rarr 0)` other ext. `= (pi)/(2) sqrt((bm)/(y))`
Time period `= pi = sqrt((m)/(y)) (sqrta + sqrtb)`
`(y x_(0)^(2))/(2a) = (yc^(2))/(2b)`
`c = x_(0) sqrt((b)/(a))`
Distance (ext to ext)
`x_(0) + c = x_(0) ((sqrtb + sqrta))/(sqrta)`