Question

To make the frequency double of a spring oscillator, we have to
A. Half of the mass.
B. Quadruple the mass.
C. Double of mass.
D. Reduce the mass to one fourth

Answer 1

Hint: Find the relation between mass and angular frequency of spring oscillator and use trial and error method to find what will be changed in mass to double the angular frequency of spring oscillator.

Formula used :
Angular frequency \[\omega = \sqrt {\dfrac{k}{m}} \]
Here, K = Spring constant and m = mass of the system

Complete step by step solution:
Frequency of a spring oscillator is defined as the number of oscillation that occurs in a unit time and it is given by,
Angular frequency \[\omega = \sqrt {\dfrac{k}{m}} \]
Where, K = Spring constant and m = mass of the system

We will use trial and error methods for the given options to find when the frequency of the oscillator is doubled. Let the mass of the system be m, then
Case 1: When the mass is halved.
New mass of the system will be \[\dfrac{m}{2}\] and,
Angular frequency will be
\[{\omega _1} = \sqrt {\dfrac{{2k}}{m}} = \sqrt 2 \sqrt {\dfrac{k}{m}} \\
\Rightarrow {\omega _1} = \sqrt 2 \omega \]
Angular frequency of the system is square root of 2 of original angular frequency.

Case 2: When the mass is quadrupled.
New mass of the system will be 4m and,
Angular frequency will be
\[{\omega _2} = \sqrt {\dfrac{k}{{4m}}} = \dfrac{1}{2}\sqrt {\dfrac{k}{m}} \\
\Rightarrow {\omega _2} = \dfrac{\omega }{2}\]
Angular frequency of the system is halved.

Case 3: When the mass is doubled.
New mass of the system will be 2m and,
Angular frequency will be,
\[{\omega _3} = \sqrt {\dfrac{k}{{2m}}} = \dfrac{1}{{\sqrt 2 }}\sqrt {\dfrac{k}{m}} \\
\Rightarrow {\omega _2} = \dfrac{\omega }{{\sqrt 2 }}\]
Angular frequency is equal to original angular frequency divided by square root of 2.

Case 4: When mass is reduced to one fourth.
New mass of the system will be \[\dfrac{m}{4}\] and,
Angular frequency will be
\[{\omega _4} = \sqrt {\dfrac{{4k}}{m}} = 2\sqrt {\dfrac{k}{m}} \\
\Rightarrow {\omega _4} = 2\omega \]
Angular frequency is doubled.
Hence, when the mass is reduced to one fourth angular frequency will be doubled.

Therefore, option D is the correct answer.

Note: In spring oscillators any change in the mass of the system will not affect the spring constant anyhow and angular frequency will change when there is any change in the mass of the system.