Question

A block of mass m is placed on ideal spring and is in equilibrium. The spring is compressed by X₀ in equilibrium. If the spring is further compressed by 2X₀ and released find the time period of oscillations:

(1) \(2\sqrt{\frac{x_0}{g}}(π+1)\)

(2) \(2π\sqrt{\frac{x_0}{g}}\)

(3) \(2\sqrt{\frac{x_0}{g}}(\frac{2π}{3}+√3)\)

(4) \(2π\sqrt{\frac{m}{k}} + \sqrt{\frac{x_0}{g}}\)

Answer 1

When spring is equilibrium then

\(F = kx\)

\(mg = kx\)

\(x = \frac{mg} k\)

when spring compressed by 2x₀

\(F = k(2x_0)-mg\)

\(ma = k2x_0 - mg\)

\(a = \frac{2kx_0}{m} - g\)

\(\because \frac{kx_0}m = g\)

\(a = \frac{kx_0}m\)      .....(1)

\(a = -w^2x\)      ......(2)

Comparing equation (1) and (2)

\(w^2 = \frac km\)

\(w = \sqrt{\frac km}\)

\(\frac{2\pi}{T} = \sqrt{\frac km}\)

\(2\pi \sqrt {\frac mk} = T\)   .....(3)

\(kx_0 = mg\)

\(\frac{x_0}g = \frac mk\)    -this put in equation (3)

\(T = 2\pi \sqrt {\frac mk}\)

\(T = 2\pi \sqrt{\frac{x_0}g}\)