A particle of mass `m` in suspended at the lower end of a thin negligible mass. The upper end of the rod is free to rotate in the plane of the page about a horizontal axis passing through the point `O`. The spring is underformed when the rod is vertical as shown in fig. If the period of oscillation of the system is `pisqrt((L)/(n))`, when it is slightly displaced from its mean position then find `n`. Take `k = (9mgL)/(l^(2))` and `g = 10m//s^(2)`.

Answer

25

Explanation

When the small angular displacement Given to rod
For small displacement
`x = l sin theta`
`kx = kl sin theta`
Torque about point `O = I_(o)alpha`
`(kx)l + mg sin theta l = mL^(2)alpha` ltbr. `alpha = ((kl^(2) + mgL)sintheta)/(mL^(2))`
For small `theta`
`alpha = -[(kl^(2) + mgL)/(mL^(2))]theta`
Time period `T = 2pisqrt((mL^(2))/(kl^(2) + mgL)) = pisqrt((L)/(25))`