Question

Which of the following mass-spring systems will have the highest frequency of vibration?
A) A spring with a $k=300N/m$ and a mass of $200g$ suspended from it
B) A spring with a $k=400N/m$ and a mass of $200g$ suspended from it
C) A spring with a $k=500N/m$ and a mass of $200g$ suspended from it
D) A spring with a $k=200N/m$ and a mass of $200g$ suspended from it

Answer 1

Before finding the frequency, we should know what a spring-mass system is. In a spring-mass system, the spring has a negligible mass and another mass is suspended from it. The suspended mass executes simple harmonic motion. We have been provided with the spring constants for different cases and we have to find the highest oscillation frequency. The point to be noted is that the suspended mass remains the same in all the cases.

Complete step by step solution:
The spring constant for the spring is given as $k$ and let the suspended mass be $m$
The time period of oscillations of the spring-mass system is given as $T=2\pi \sqrt{\dfrac{m}{k}}$ where the meaning of the symbols have been given above.

The frequency of any system can be given as the reciprocal of its time period, that is $f=\dfrac{1}{T}$
Substituting the value of the time period as given above, we get Frequency of oscillations of the spring-mass system $(f)=\dfrac{1}{2\pi }\times \sqrt{\dfrac{k}{m}}$.

Since we have to find the system having the highest frequency among all the given options and the suspended mass does not change, we can say that $f\propto \sqrt{k}$ since $\dfrac{1}{2\pi }\times \sqrt{\dfrac{1}{m}}$ will give us the same value for all options and can be taken as a constant
Since we now know that $f\propto \sqrt{k}$ , it means that the system having the largest value of the spring constant will possess the largest frequency of oscillation.

Hence we can say that for spring with $k=500N/m$ and a mass of $200g$ suspended from it, the frequency of oscillation will have the largest value.

Therefore, option (C) is the correct answer.

Note: In the given question, we were able to directly find the answer by comparing the spring constants only because the mass had a constant value in all options. You cannot use this method if the suspended mass has a different value for a different option. In that case, you will have to substitute the values of the spring constant and the suspended mass, find the value of frequency for each case and then compare the results.