Question

Two springs of force constants K and 2K are connected to a mass as shown below. The frequency of oscillation of the mass is

A. \[\dfrac{1}{{2\pi }}\sqrt {\dfrac{K}{m}} \]
B. \[\dfrac{1}{{2\pi }}\sqrt {\dfrac{{2K}}{m}} \]
C. \[\dfrac{1}{{2\pi }}\sqrt {\dfrac{{3K}}{m}} \]
D. \[\dfrac{1}{{2\pi }}\sqrt {\dfrac{m}{K}} \]

Answer 1

Hint: Frequency of oscillation directly proportional to the square root of spring constant and in case of combination of springs the effective spring constant for the system is taken under consideration.

Formula used :
\[n = \dfrac{1}{{2\pi }}\sqrt {\dfrac{K}{m}} \]
Here, n = Frequency of oscillation, K = Spring constant and m = Mass

Complete step by step solution:
Frequency of oscillation is defined as the number of oscillation that took place in a unit of time i.e. second and is mathematically given by the formula,
\[n = \dfrac{1}{{2\pi }}\sqrt {\dfrac{K}{m}} \]
Where, Here, n = Frequency of oscillation, K = Spring constant and m = Mass

For the given spring mass system spring constant of individual spring is K and 2K respectively and the frequency of oscillation the system will be,
\[n = \dfrac{1}{{2\pi }}\sqrt {\dfrac{{{K_{eff}}}}{m}} \,\]
Where,\[{K_{eff}} = \]Effective constant of the system.

Effective constant for the given system of springs constant K and 2K will be the sum of both constants.
\[{K_{eff}} = K + 2K\, = 3K\]
So the frequency of oscillation of the system will be,
 \[n = \dfrac{1}{{2\pi }}\sqrt {\dfrac{{3K}}{m}} \,\]

Therefore, option C is the correct answer.

Note: When two or more springs are connected end to end then they are said to be in series combination and in parallel combination springs are connected side by side.