Figure. (a) shows a spring of force constant k fixed at one end and carrying a mass m at the other end placed on a horizontal frictionless surface. The spring is stretched by a force F. Figure. (b) shows the same spring with both endsf free and a mass m fixed at each free end Each of the spring is streched by the same force F. The mass in case (a) and the masses in case (b) are then released. Which of the following statements are true?

Options

A. While oscillating the maximum extension of the spring is more in case (a) than in case (b).
B. The maximum extension of the spring is same in both cases.
C. The time period of oscillation is the same in both cases.
D. The time period of oscillation in case (a) is `sqrt2` time that in case (b).

Answer

B, D

Explanation

The maximum extension `x` produced in the spring in case (a) is given by
`F=kx` or `x=(F)/(k)`
The time period of oscillation is
`T=2pisqrt((mass)/("force constant"))=2pisqrt((m)/(k))`
In case (a) one end A of the spring is fixed to the wall when a force F is applied to the free and B in the direction, the spring is stretched exerting a force on the wall which in turn exerts an equal and opposite reaction force on the spring, as a result of which every coil of the psring is elogated producing a total extension x.
In case (b) both ends of the spring are free equal forces area applied at ends. by application of forces both the cases are same.
Thus, the maximum extension produced in the spring in cases (a) and (b) is the same. In case (b) the midpoint of spring wil not move, we can say the blocks are connected with the springs whose lengths are half the original length of spring. Now, the force constant of half the spring is twice of complete spring. In case (b) the force constant `=2k`. Hence, the time period of oscillation will be
`T'=2pisqrt((m)/(2k))`
`(T')/(T)=sqrt2`
hence, the correct choices are (b) and (d).