Natural Frequency of a Spring Mass System by Equilibrium Method

Q. A single degree of freedom system having mass 1 kg and stiffness $10\frac{kN}{m}$ initially at rest is subject to an impulse force of magnitude 5kN for ${10}^{-4}$ seconds. The amplitude in mm of the resulting free vibration is

1. 0.5
2. 1.0
3. 5.0
4. 10.0

Q. A small pan suspended from a helical spring as shown in the figure, empty has an observed period of vibration $T_0$. When carrying a known weight P, the observed period of vibration is $T_1$ and when carrying an unknown weight W (without P), the observed period is $T_2$. Find the magnitude of the weight W.

1. $\left(\frac{T_2+T_1}{2T_0}\right)P$
2. $\left(\frac{T_2+T_1}{T_1+T_0}\right)P$
3. $\left(\frac{T_2+T_1}{T_1+T_0}\right)P$
4. $\left(\frac{T_2^2+T_0^2}{T_1^2+T_0^2}\right)P$

Q. A simple mass-spring oscillatory system consists of a mass $m$, suspended from a spring of stiffness $k$. Considering $z$ as the displacement of the system at any time $t$, the equation of motion of the free vibration of the system is $m\ddot{z}+kz=0$. The natural frequency of the system is

1. $\sqrt{\frac{k}{m}}$
2. $\sqrt{\frac{m}{k}}$
3. $\frac{k}{m}$
4. $\frac{m}{k}$