What is the correct formula I should use? (measuring g using a swinging mass)

[MJNZ1]

Homework Statement

What is the acceleration due to gravity?

Relevant Equations

Formula 1 F = (4pi^2*m*r)/(T^2)
Formula 2 g exp. = (4pi^2*m*r)/(M*T^2) or g = (4pi^2*m*r)/(M*T) ?

Hi,

This is my first post, I hope I have done it right.

The top mass is represented as m and the weight hanger M.The image below is my attempt with the first formula 2. I have done it in grams because when I did it in kg's it was not anywhere near 9.81ms^-2...I know it is wrong :)

g exp. = (4pi^2*m*r)/(M*T^2) ...in kg's: (4pi^2*0.02*0.55)/(0.1*4.63^2) = 0.938ms-^2

When the teacher was asked by another student for further help understanding, the second formula 2 was given.

This is my attempt at solving for gravity from the example below in kg's: (4pi^2*0.02*.55)/(0.1*4.63) = 0.938ms^-2

g = (4pi^2*m*r)/(M*T)

 

[Delta2]

I am not very familiar with this type of experiment to measure g, but IF I have understood correctly the experiment, i think that the correct formula is $$g=\frac{F_c}{M}=\frac{m\omega^2 r}{M}$$ which becomes (after replacing $\omega=\frac{2\pi}{T}$, $$g=\frac{4\pi^2mr}{MT^2}$$

 

[MJNZ1]

Thank you for your reply.

What is w? tried it as Newtons but it didn't work out so well, unless I've entered it in wrong?

 

[haruspex]

In the screenshot of the spreadsheet, the cell formula has a 20 where there should be a pi(). This makes the computed values in columns F and K nonsense.
In the hand calculation, (4pi^2*0.02*.55)/(0.1*4.63) = 0.938ms^-2, the 4.63 should be squared, leading to an even lower value.

Where do these data come from? Did you conduct the experiment? Periods of the order of 4 or 5 seconds seem ridiculously long. Usually one counts a number of revolutions, so maybe somebody forgot to divide by the count.

How was it arranged that it was steady state?

In reality, the string would never be horizontal. This leads to a slightly different equation $g=\frac{4\pi^2mr}{\sqrt{M^2-m^2}T^2}$, but this only makes a small difference here.

 

[Delta2]

Where do these data come from? Did you conduct the experiment? Periods of the order of 4 or 5 seconds seem ridiculously long. Usually one counts a number of revolutions, so maybe somebody forgot to divide by the count.

How was it arranged that it was steady state?

I agree strongly with you here.

 

[MJNZ1]

Oh, of course. Yes, I did, it was before our two-week break, so I had forgotten and didn't note down the number of revolutions. The 20 was for F1 because I mistook pi for F1 and hadn't changed.

Thank you both for your help! ...it must be getting late

 

[Delta2]

I have my doubts regarding the measurement of period. As @haruspex said, periods of 4-5 seconds seem too big for this experiment. Can you tell us how did you measure the period?

 

[haruspex]

I have my doubts regarding the measurement of period. As @haruspex said, periods of 4-5 seconds seem too big for this experiment. Can you tell us how did you measure the period?

In post #6, @MJNZ1 indicates it was for some number of revolutions, but, sadly, has forgotten how many.
Of course, I am not going to suggest working back from 9.81 and the other data to find out...

 

[Delta2]

In post #6, @MJNZ1 indicates it was for some number of revolutions, but, sadly, has forgotten how many.
Of course, I am not going to suggest working back from 9.81 and the other data to find out...

Oh i see well working back from the data i found out that if the number of revolutions is 7-8 then we end up with 9.81 using the formula i wrote in post #2.

 

[MJNZ1]

I think it was 10. This is what I have gotten so far, I will fix the rest tomorrow once I've had some sleep.

There was a slight gradient in the string, but it wasn't part of the criteria to calculate for this. I'm just in a basic pre-engineering course because I have been out of school for 16yrs, I guess they just want to keep labs simple at the moment. :)

 

[haruspex]

The slope correction would lift the value of g by about 2%, which gets you usefully closer to the correct value.
What mean do you get?
How are you calculating the confidence interval?

 

[MJNZ1]

Our teacher set us up a calculator in excel, I haven't done a mean...I desperately need some sleep, lol.

 

[haruspex]

Our teacher set us up a calculator in excel, I haven't done a mean...I desperately need some sleep, lol.

Looks like the 9.5 number is the mean. The +2% correction would get you to 9.7, so definitely more creditable.
The quoted confidence interval seems to be a bit over two standard errors of the mean. One standard error would be stdev/√(n).

 

[MJNZ1]

Looks like the 9.5 number is the mean. The +2% correction would get you to 9.7, so definitely more creditable.
The quoted confidence interval seems to be a bit over two standard errors of the mean. One standard error would be stdev/√(n).

Thanks, I will do the taught method first and if I get enough time I will also add the equation that you gave in question 4. I will fix up the confidence interval, thanks for checking that. :)

 

[haruspex]

Thanks, I will do the taught method first and if I get enough time I will also add the equation that you gave in question 4. I will fix up the confidence interval, thanks for checking that. :)

No, no need to change anything re the confidence interval. That depends on the level of confidence specified, and two standard deviations may well be what was chosen.