The relationship between time taken per oscillation and mass

[WaterMelllon1]

In an experiment, a ruler is connected to the table and some weights are bounded to one end of the ruler. The ruler is then flicked and the time taken per oscillation is measured.

I have plotted a graph with the data I have collected, with the mass on the y-axis and time on x-axis. The graph produced appears to be a curve. I have tried altering the values on the x-axis; I have squared it, 1 over the square of it, square rooted it, and I found that the graph becomes linear when the values are squared. So the mass should be proportional to 1 over the square of the time taken.

I have tried finding out a mathematical relationship for this, but I am not sure if this is correct or not.

Well, if we make w=angular velocity, then w=θ/t, with θ being angular displacement and t being the time period. Since θ belongs in a circle, then it is safe to say that w=2π/t (?)

Also, if the force of an oscillation is proportional to -displacement (x), then it is true to say that F=-kx, with k being a constant.

Since F also = ma, then ma=-kx.

According the the simple harmonic wave equation for acceleration is a=-xw²sinwt. Since the formula for displacement(x) = x sinwt and a=-w²(x sinwt), then a=-xw²

So ma=-kx will become m(-xw²)=-kx, then using some algebra, m=k/w². Since w=2π/t, then m=kT²/4π².

Since k/4π² is a constant, I can ignore that and say m is proportional to t².

Is my reasoning true? I feel like I am wrong in quite a few spots.

Thanks for helping

 

[Gokul43201]

Yes, this is essentially correct, but you take a few confusing steps because of a poor choice of notation. For instance, you use x to represent displacement as well as the amplitude (or maximum displacement). You should use different symbols for these - it will help keep things clear for you as well.

You write: x = x sin(wt), which looks nonsensical.

Better would be something like: x = A sin (wt), or x = x₀ sin (wt).

Also, next time, a question like this is better suited for the Intro Physics section.

 

[WaterMelllon1]

Thank you.

 

[WaterMelllon3]

Yes thank you very much! It is so helpful!

 

[Nickie]

me out!
Your reasoning is mostly correct. The relationship between mass and time taken per oscillation can be described by the equation m = kT^2/4π^2, where m is the mass, T is the time taken per oscillation, and k is a constant. This is known as the period-mass relationship and it is derived from the equation of simple harmonic motion, F = -kx, where F is the force, k is the spring constant, and x is the displacement.

In your experiment, you have found that the graph of mass vs. time taken per oscillation is a curve, and when the time values are squared, the graph becomes linear. This is because the period-mass relationship is a quadratic equation, so when you square the time values, you are essentially plotting the equation in a linear form.

Your reasoning for using the equation a = -xw^2 is also correct. This is the equation for acceleration in simple harmonic motion, where w is the angular velocity. However, you have made a small mistake in your algebra. It should be ma = -kx, not ma = -xw^2. This is because the force of an oscillation is proportional to the displacement, not the acceleration.

Overall, your reasoning is correct and you have a good understanding of the relationship between mass and time taken per oscillation in simple harmonic motion. Keep up the good work!