Calculating Spring Constant From Plot of T^2 versus r^2

[Physics Dad]

Homework Statement

Hi,

I recently performed a lab experiment for calculating the moment of inertia and spring constant of a couple of equal masses on a steel rod. I had to do this experiment at the exact same time as performing another experiment so I was unable to perform any calculations in the lab, just simply gather data.

The first part of the experiment requires the calculation of the torsional spring constant (k) using a plot of T² v r² and as I was unable to perform calculations in the lab, I was hoping someone could provide a little guidance here.

Homework Equations

T²=4π²(I/k)
r²=I/2m
y=Δx+c (where Δis the gradient (wrong symbol, I know))

(T is the period of one single oscillation, r is the radius of the masses from the centre of mass)

The Attempt at a Solution

I know that T² = 4π²(I/k) and that r² = I/2m (because there were two equally spaced masses on a thin rod of negligible mass). If I then plot T² v r² in JLineFit and equate this to the the equation of a straight line, I should find that T² = Δr²+c.

Subbing in my values for T² and r² and solving for k, I get the equation to be:
4π²(I/k) = (ΔI/2m)+c

so if I start by multiplying through by k I get

4π²I=k((ΔI/2m)+c)

so

4π²I/((ΔI/2m)+c)=k

so

k = 4π²/((Δ/2m)+c)

All I really want to know is if my rearranging is correct (do the I's cancel?). I get very confused here!

Thanks

 

[kuruman]

y=Δx+c

What is the relevance of this equation to your experiment? What do the symbols mean in terms of something that you measured?

T2 = Δr2+c.

How do you use the symbol Δ? Usually it means a change.

 

[Physics Dad]

it is the equation of a straight line (y=mx+c) I should have used ∇.

T is the time period of one single oscillation, r is the radius of the mass.

Thanks

 

[scottdave]

OK, so since c is an arbitrary constant (to be determined from the data), then k*c can also be just an arbitrary constant. When you divided through by I to cancel, you would have an c/I, which is also an arbitrary constant, as well. This could simplify some things for you when you had 4π²I=k((ΔI/2m)+c) = =k*(ΔI/2m) + (k*c), where k*c, or even k*c/I can be just an arbitrary constant (to be determined).

 

[kuruman]

To make things simple: You have $T^2=4 \pi^2 I/k$ and $I = 2mr^2$. If you replace the $I$ in the first equation, you get
$T^2=8 \pi^2 mr^2/k$. Now let $y=T^2$, $slope=8 \pi^2 m/k$ and $r^2 = x$. Then you have the standard straight line equation $y=slope*x$. I used "slope" instead of m in order not to confuse the "slope" m with the "mass" m. You can add an intercept $c$, if you wish, but you need to explain what it means to have some non-zero period when r = 0. Once you get a number for the slope, you can solve $slope=8 \pi^2 m/k$ to find $k$. This process is known as linearization.