Question

A spring mass system oscillates with a time period $T_1$ when a certain mass is attached to the spring. When a different mass is attached after removing the first mass the time period becomes $T_2$. When both the masses are attached to the free end of the spring with the other end fixed, the time period of oscillation is :

• A. $\sqrt{T_1{T}_2}$
• B. $T_1+T_2$
• C. $\frac{T_1+T_2}{2}$
• D. $\sqrt{T_1^2+T_2^2}$

Answer 1

Answer: (D)

$\sqrt{T_1^2+T_2^2}$

$T_1=2\pi \sqrt{\frac{m_1}{a}},m_1=\frac{T_1^{2}a}{4\pi 2}$
$T_2=2\pi \sqrt{\frac{m_2}{a}},m_2=\frac{T_2^{2}a}{4\pi 2}$
$T=2\pi \sqrt{\frac{m_1+m_2}{a}}$
$\therefore m_1+m_2=(t_1^2+T_2^2)\frac{a}{4\pi 2}$
$\therefore T=\sqrt{T_1^2+T_2^2}$