
How does the period of an oscillation in a spring-mass system depend on mass and amplitude? How can I express this graphically for period as a function of mass, period as a function of the square root of the mass, and period as a function of amplitude?
Answer
446.4k+ views
Hint: In order to answer the following question, we will be describing the most basic spring-mass system. We will derive the equation of the time period of oscillations of such a system. Finally, we will represent the graph of time period with different types of variables as asked in the question.
Formula used:
Time period of oscillation:
Where is the mass and is the spring constant.
Complete step by step answer:
First of all, we will draw a simple diagram of a spring-mass system.
As we can see from the figure, a mass is suspended with the help of a spring having a spring constant of ( ). If the spring is stretched and then released, the spring-mass system will oscillate. The formulae for time period for such oscillation is given by
If we square both sides of the above equation, the equation becomes
The above equation implies that the time period of oscillation is directly proportional to the square root of mass. It also states that the time period is independent of the amplitude.Therefore, if we plot a graph of vs , we will have a square root graph
Now, if we plot a graph of vs , then we get a linear graph with a gradient equal to
Lastly, the graph of vs (amplitude) will represent a horizontal line as time period is independent of amplitude.
Note:It is very important to keep in mind that the graph of vs will give you a horizontal line because time period is independent of amplitude and will remain constant with change in amplitude. Now, if the spring is overstretched or if it jumps, then this dependence may vary and the graph will not be a horizontal line.
Formula used:
Time period of oscillation:
Where
Complete step by step answer:
First of all, we will draw a simple diagram of a spring-mass system.

As we can see from the figure, a mass
If we square both sides of the above equation, the equation becomes
The above equation implies that the time period of oscillation is directly proportional to the square root of mass. It also states that the time period is independent of the amplitude.Therefore, if we plot a graph of

Now, if we plot a graph of

Lastly, the graph of

Note:It is very important to keep in mind that the graph of
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