Determining a centripetal force proportionality equation

[ova5676]

Homework Statement

In a lab, we essentially used an apparatus similar to this:

In three experiments with several trials, we varied three things: centripetal force (the mass of the bottom), radius (of the circle by adjusting the alligator clip), and mass (rubber stopper). The trials were varied as well (increasing radius, increasing bottom mass, increasing rubber stopper mass). So, we ended up with 3 graphs for each experiment where each point was a trial. My question is:

1) How do I combine those proportionality into one equation in terms of Fc?

2) How can I find a proportionality constant

3) How can I find the percent error for k?

Homework Equations

According to my graphs, the proportionality constants were T ∝ r, T ∝ 1/Fc and T ∝ m^2.

The Attempt at a Solution

1) That means T = r, T = 1/Fc, and T = m^2, correct?

So r = 1/Fc = m^2. How would I get that to be Fc = etc. etc.?

2) Using whatever equation is produced from 1), I would put k after Fc. So, say Fc = m^2 * r / T, it would be Fc = km^2*r/T.

How would I find k though? What values do I substitute? I have three different experiments with several trials within each one that vary?

3) I don't know how to find % error for k? Would that be slope as the theoretical and the actual is the calculate one from 2)?

 

[mfb]

1) That means T = r, T = 1/Fc, and T = m^2, correct?

If the units don't match it cannot be correct. A time is not a distance, for example.
If a quantity is proportional to x and proportional to an independent y then it's proportional to the product of x and y:

T ∝ r m²/Fc

Assuming T is the revolution period your relations are wrong, however, the left side should always be T² and in the last case T² ∝ m

T² ∝ r m/Fc

Introducing a proportionality constant c:

T² = k r m/Fc

And solving for k:

k = T² Fc / (r m)
You can calculate that value for every measurement and find k. Ideally you get very similar values for all individual data points. The spread of the measurements gives an idea how good the measurement is.

 

[nagham]

how do I find the value of t while it is not given?
in a case like this I have two missing values!

 

[haruspex]

how do I find the value of t while it is not given?
in a case like this I have two missing values!

How are you defining t? Post #1 refers to T, which is presumably the rotation period and would have been measured as part of the experiment.

 

[kuruman]

In three experiments with several trials, we varied three things: centripetal force (the mass of the bottom), radius (of the circle by adjusting the alligator clip), and mass (rubber stopper). The trials were varied as well (increasing radius, increasing bottom mass, increasing rubber stopper mass). So, we ended up with 3 graphs for each experiment where each point was a trial.

After a trial was over, you recorded the value of whichever of the three things you varied and the fixed values of the other two. Is that all you recorded? If you recorded the period, which is the dependent variable, note that you have three measured independent variables. I think what you are expected to do is combine all three into a single independent variable. You can do that if you find an expression for $T^2$ and put it on the left-hand side. The right-side can only have the three independent variables and constants like $g$ and $\pi$. I will not tell you what the equation looks like because this is a homework thread and you have to find the equation on your own. A good start in that direction is provided by @mfb, however the expression contains F_c which is not one of the three independent variables.

The final equation will look like $$T^2=k~X$$where $k$ is the proportionality constant that (I think) you are seeking and $X$ is the particular combination of hanging mass $m_h$, cork mass $m_c$ and radius $R$ that appears in your derived equation for $T^2$.

Note that if you have taken $n_1$ data points varying $m_h$, $n_2$ data points varying $m_c$ and $n_3$ data points varying $R$, the total number of data points will be $N=n_1\times n_2 \times n_3.$ Plot all $N$ of them on a single graph of $T^2$ vs. $X$ and figure out the slope $k$. Then see if it agrees within experimental error (error propagation is needed for that) with the constant value that is predicted by the equation that you derived for $T^2.$

 

[haruspex]

After a trial was over, you recorded the value of whichever of the three things you varied and the fixed values of the other two. Is that all you recorded?

Post #1 is 11 years old. Post #3 is the current question.

 

[kuruman]

Post #1 is 11 years old. Post #3 is the current question.

Right you are. I didn't think a mentor would reply to an eleven year old question so I didn't check the date of the original post. I shall have to be more careful.

 

[haruspex]

Right you are. I didn't think a mentor would reply to an eleven year old question so I didn't check the date of the original post. I shall have to be more careful.

Thank you for the promotion.

 

[kuruman]

Considering all the pro bono work that mentors have to do, I am not sure it's a promotion - superpowers notwithstanding. Anyway, I was referring to the fairly recent post #2 by @mfb that started this ball rolling.

 

[mfb]

This was part of an effort to reduce the number of unanswered but popular threads (in search engines).

@nagham T would be measured in this experiment. If you have a question about a different experiment, please start your own thread.