# The physical meaning of a slope

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1. May 27, 2017

### ddddd28

1. The problem statement, all variables and given/known data
Yesterday, our physics class had a laboratory examination, in which we used a protractor as a "weight".
We hung some paperclips and checked the angle.

Then, they asked about the physical meaning of the slope of the graph: tan( angle) as a function of n( number of paperclips). The relation is linear.
The slope has no dimensions, as you can see.
2. Relevant equations

3. The attempt at a solution
Unsatisfied with the teacher's answer, which had mathematical meaning, I persisted to find the physical meaning. A slope can not be just a random value, it must be consisted of some constants.
So, my intuition is that the slope is the ratio between the mass of a single paperclip and the mass of the counter side of the protractor.
This is due to the fact that the bigger the mass of the paperclip is, the greater tan( angle) would be, so the mass must appear in the slope.
I tried to prove it, or to find a general relation between the tan and n, but with no success.
I would like to hear your ideas of what the slope could be.

Thanks.

2. May 27, 2017

### lampCable

Have you considered the torque about point B?

3. May 27, 2017

### ddddd28

Actually, I have. But, I couldn't find the relation.

4. May 27, 2017

### zwierz

think about protractor's center of mass

5. May 27, 2017

### Ray Vickson

What was the teacher's answer? Why are you unsatisfied with it? It is one thing to be unsatisfied with something that is wrong, but quite another to be unsatisfied because it is "mathematical". Welcome to the world of Physics, where analyses and explanations are commonly mathematical, just because the laws of Physics are mathematical in nature. I'm not sure I know the difference between "physical meaning" and "mathematical meaning".

On the other hand, there is certainly nothing wrong with seeking alternative explanations or with expressing things in another way. That is a good way of learning and expanding your knowledge base. So: good luck, but don't be shocked or dismayed if that sometimes does not work our for you.

6. May 27, 2017

### ddddd28

The "right" answer according the teacher was that the meaning of the slope is the ratio between the difference of the tan angle and the difference of n, which is true, but in the same time, doesn't contribute to your understanding. For example, the slope of x(t) is the ratio between the change of distance and the time - this is mathematical meaning. The physics would say "velocity".
I am dissastified not because it might be wrong, but because the question dealt with the physical meaning.

7. May 27, 2017

### zwierz

you have to get used to that the many physical effects are described by means of math. and have no direct physical explanation. Only very trivial effects have such an explanation.
For example

8. May 27, 2017

### vela

Staff Emeritus
What did you get? Your intuition above gave you the right answer, but you should be able to derive it mathematically. It's pretty straightforward.

9. May 27, 2017

### Ray Vickson

You and your teacher seem to have different interpretations of the word "meaning". Your teacher gives an accurate (but in my opinion unenlightening) explanation of what slope is, but you (and I agree with you) seem to want more of an explanation of why or how it arises---for example, its relation to the masses of the paper clips, etc. I would not dismiss the teacher's explanation as unsatisfactory just because it looks mathematical, but rather because it does not deal with the substantive issue in the experiment. Substantive explanations might be pretty mathematical, like it or not.

BTW: I disagree with your explanation of the "mathematical" vs. "physical" meaning of rate of change of distance with respect to time. Calling something "velocity" explains nothing---it is just a word. Explaining the meaning of that word in mathematical terms is crucial to the definition; after that has been done, we can use the word "velocity" as a kind of shorthand that bypasses the need for constant re-explanation of the math in every case of interest.

10. May 27, 2017

### ddddd28

Thanks, for all of your responses.
Whether the teacher's answer is legitimate or not, I think most of you missed the intention of the original question:" how could I find which constants the slope consists of?"
Vela argued that my intuition is right, but not knowing moments well enough, I couldn't derive it easily. (I tried also to approach the problem with double circular motion, but finished with a mess)

11. May 27, 2017

### Ray Vickson

PF rules forbid us from "teaching" you; we are allowed to offer hints and criticisms, as well as recommendations for further reading, etc.

12. May 27, 2017

### ddddd28

Yes, it is known to me...
so maybe hints will be helpful

13. May 27, 2017

### vela

Staff Emeritus
I guess I shouldn't have said it was the right answer, but you're right that it depends on the ratio of the masses.

You actually have been given hints already: consider torques and the protractor's center of mass. Take a stab at it and tell us what you come up with.

14. May 28, 2017

### ddddd28

This is my attemp:

M- the mass of paperclips
m-the counter mass
l- the lengh of the rope
r- the half lengh of the protractor

The system will rest when the torques are zero.
τ1=τ2
√(r^2+l^2-2rlcos β) * Mg* sinγ= mg* r *sin β
sin γ= r*√(r^2+l^2-2rlcos β) /sin β by sine theorem
therefore:
(r^2+l^2-2rlcos β)* M /sin β = m sin β
α=180-β between the r and the vertical axis
(r^2+l^2-2rlcos β)*M = m sin^2 β
.
.
.
r^2*M+l^2*M-m = -cos^2 α *m -2rlcos α *M
Here, I tried to solve a quadratic equation with refernce to cos α, and divide √1-cos^2 α by the result to get tanα, but as you might expect, it is too complex...
Have I made a wrong assumption?

15. May 28, 2017

### vela

Staff Emeritus
Good start.

That should be $\frac{\sin\gamma}{r} = \frac{\sin \beta}{\sqrt{r^2+l^2-2rl\cos\beta}}$. In your expression, the lefthand side is unitless where as the righthand side has units of length squared, so it can't be right. Anyway, if you work that last step out correctly, you'd get some cancellation and end up with $Mgr\sin\beta$. You could have gotten to this expression directly from using the fact that the torque is equal to the force times the distance from the pivot point to the line of action of the force.

There's a problem with the other side of the equation. (By counter mass, I assume you're referring to the mass of the protractor.) Imagine the protractor is horizontal and there are no paper clips so that any net torque is only due to the protractor's mass. The way you defined the angles, you'd have $\beta = 90^\circ$ and $mgr\sin\beta \ne 0$. In other words, there'd be a torque on the protractor causing it to rotate. Is that what you'd expect? Where is the center of mass of the protractor?

16. May 28, 2017

### haruspex

I do not understand the diagram in post 14. In the OP you wrote
In the diagram in the OP, where is the pivot? I assumed it was at B, but if so, what is that ring suspended from point B? Is that supposed to represent the weight of the protractor? I suggest it would be a bit further to the left, otherwise no balance would be achieved.