Understanding the Pendulum Equation: How Do We Go from Slope to Equation?

[porter519]

Homework Statement

ok well we are doing a lab and we have to figure out the pendulum equation. I know that I can look it up online but she wants us to figure it out for ourselves and have the work. What we did was we set up the pendulum at different lengths and recorded how long it took for ten swings and then divided that by ten, which in turn was how long 1 period was.we then plotted period vs length of the string and found a slope. now here is where I am stuck. what exactly is the slope supposed to represent? and how do I go from the slope to the equation?

Homework Equations

This is an equation that she gave us T=K(sqrtL) where T= period, K=slope, and L= length of the string... Idk if this'll help tho

The Attempt at a Solution

 

[cristo]

What slope did you find when you plotted the graph? Are you sure you did not plot a graph of T² vs. L ?

If you did not do the latter, then I suggest you do it. If you did, do you know the equation of a straight line?

 

[rock.freak667]

Well you should have plotted (Period)^2 against Length to get a straight line...

ok well let's use units to help us

$$T^2=kl$$ where k is the gradient of a graph of $$T^2$$ vs $$l$$

and therefore $$k=\frac{T^2}{l}$$ meaning that the units of k is $$\frac{s^2}{m}$$ or better put $$s^2m^{-1}$$

now let's for a moment just put the units of k into a way that looks better(usually with m as i +ve power),to do this we would have to put 1 over those units. Right?

giving $$\frac{1}{ms^2}$$ ...Ah...what's this $$ms^2$$ you know what these units represent...and the only thing with such units that affects your pendulum is acceleration due to gravity...and so and so your gradient is actually a ratio of $$Some Number,P : g$$ meaning that

$$T^2=\frac{P}{g}l$$ now using your value for k and a value for g...find that number P and see if it looks closely to the value of $$4\pi^2$$

 

[porter519]

oh wow thanks ^_^ lol that was a lot of help