A weightless spring which has a force constant k oscillates with frequency n when a mass m is suspended from it. The spring is cut into two equal halves and a mass 2m is suspended from it. The frequency of oscillation will now become
A. n
B. \[m\left( {g + \sqrt {\dfrac{{{\Pi ^2}}}{2}gh} } \right)\]
C. \[\dfrac{n}{{\sqrt 2 }}\]
D. \[n{(2)^{\dfrac{1}{2}}}\]

Hint:Frequency of oscillation of spring is defined as a number of complete cycles per second and it is inversely proportional to the time period of oscillation.

Formula used:
\[n = \dfrac{1}{T} = \dfrac{\omega }{{2\Pi }}\] and \[\omega = \sqrt {\dfrac{k}{m}} \]
Where,
T= Time period of oscillation,\[\omega \] = Angular frequency of oscillation
k= Spring constant and m = Mass suspended to spring

Complete step by step solution:
Given here is a weightless spring of constant k and frequency n when mass m is suspended from it, we have to find frequency of oscillation when spring is cut in two equal lengths and mass 2m is suspended from it.

The frequency of oscillation n is given by the equation,
\[n = \dfrac{\omega }{{2\Pi }}\]

Using \[\omega = \sqrt {\dfrac{k}{m}} \] in above equation frequency n can be written as,
\[n = \dfrac{1}{{2\Pi }}\sqrt {\dfrac{k}{m}} \,.........(1)\]

When the spring is cut in two equal halves spring constant k’ for each half will be 2k, let the new oscillation frequency be n’ when mass 2m is suspended from the spring.
Then, \[n' = \dfrac{1}{{2\Pi }}\sqrt {\dfrac{{k'}}{{2m}}} \] or it can be simplified as,
\[n' = \dfrac{1}{{2\Pi }}\sqrt {\dfrac{{2k}}{{2m}}} \,.........(2)\]

Using equations (1) and (2) ratio of oscillation frequencies n and n’ will be
\[\dfrac{{n'}}{n} = \dfrac{1}{{2\Pi }}\sqrt {\dfrac{{2k}}{{2m}}} \, \times 2\Pi \sqrt {\dfrac{m}{k}} \]

Further simplifying the above equation we get,
\[\dfrac{{n'}}{n} = \sqrt {\dfrac{k}{m} \times \dfrac{m}{k}} \, \Rightarrow n' = n\]

Hence, frequency oscillation when the spring is cut into two halves and mass 2m is suspended from it will be n.

Therefore, option A is the correct option.

Note:When a body performs oscillation it has both linear and angular displacement and the angular displacement of the body is called its angular frequency, angular frequency is the scalar measurement of the angular displacement of the body during oscillation.