Question

An object of mass M oscillates on the end of a spring. To double the period, replace the object with one of mass: (a) 2M. (b)M/2. (c) 4M. (d)M/4. (e) None of the above.

Answer 1

Answer:

(c) 4M

Explanation:

The system is a loaded spring. The period of a loaded spring is given by

$T = 2\pi\sqrt{\dfrac{m}{k}}$

m is the mass and k is the spring constant.

It follows that, since k is constant,

$T\propto\sqrt{m}$

$\dfrac{T}{\sqrt{m}} = C$ where C  represents a constant.

$\dfrac{T_1}{\sqrt{m_1}} = \dfrac{T_2}{\sqrt{m_2}}$

$m_2 = m_1\left(\dfrac{T_2}{T_1}\right)^2$

When the period is doubled, $T_2 = 2T_1$.

$m_2 = m_1\left(\dfrac{2T_1}{T_1}\right)^2 = 4m_1$

Hence, the mass is replaced by 4M.