# Differentiation: small increments

1. Aug 14, 2015

### DevonZA

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

1. The centripetal acceleration of a particle moving in a circle is $a = \frac{v^2}{r}$, where v is the velocity and r is the radius of the circle. Approximate the maximum percentage error in the calculation of the acceleration if the error in experimental measurements of v is ± 0,2% and in r is ± 0,8%.

2. Differentiation: small increments

3. See attached.

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2. Aug 14, 2015

### Qwertywerty

You know , there is a simple way -
Δa/a = 2Δv/v + Δr/r , for small Δv and Δr .
The reason the two quantities on the right are added is because they both have a possible plus - minus error .

Hope this helps .

3. Aug 14, 2015

### DevonZA

$\frac {Δa}{a} = \frac {2(0.2-(-0.2))}{0.2} + \frac {(0.8-(-0.8))}{0.8}$

4. Aug 14, 2015

### Qwertywerty

Wrong .
Okay , perhaps I should have mentioned this earlier - Δx ( x is just some random variable ) represents uncertainty / error in the measurements of x .

So , now , what is Δv , and what is Δr ?

5. Aug 14, 2015

### DevonZA

No, I'm lost sorry?

6. Aug 14, 2015

### Qwertywerty

These values seem like Δv/v and ...

7. Aug 14, 2015

### DevonZA

...Δr/r. They are the change in v and r because both can increase or decrease but I still don't understand what you are getting at and surely Δv = +-0.2 and Δr = +-0.8? Why put Δv over v and Δr over r?

8. Aug 14, 2015

### Qwertywerty

Over here ? -
This is derived from -
Suppose error in a is da , and in some random variable x is dx .
Then , error in a w.r.t - x is
da/dx .
Also , a = v2/r .
Hence differentiating w.r.t x on both sides ,
da/dx = 2vrdv/dx - v2dr/r2.dx - Chain rule
⇒da = 2vrdv - (v/r)2 . - Cancelling dx
⇒da/a = 2dv/v - dr/r - Dividing by a

But error is plus - minus , and hence it would become
da/a = 2dv/v + dr/r .

I hope that was your doubt .

9. Aug 14, 2015

### DevonZA

I am so confused but thank you for attempting to get me to understand

10. Aug 14, 2015

### Qwertywerty

What part ?

11. Aug 14, 2015

### DevonZA

All of it to be honest. Maybe it is the way it is written that is confusing me

12. Aug 14, 2015

### HallsofIvy

Staff Emeritus
Whoever gave you this problem expects you to know that an approximation to the error in a is $da= \frac{2v}{r}dv- \frac{v^2}{r^2}dr$ where dv and dr are the respective errors.

13. Aug 16, 2015

### DevonZA

Final answer attached. Thanks to all who helped.

#### Attached Files:

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14. Aug 16, 2015

### Ray Vickson

Please type out your solution; I cannot open the pdf file on my new i-Phone, and you should not make me install an App just to read your work.

15. Aug 16, 2015

### Staff: Mentor

I agree with Ray. Posting hand-written work, which is often difficult to read, makes it harder for us as helpers to give you assistance. For one thing, if we spot a mistake, we have to provide context to indicate which line is in error.

For another thing, having the work in a separate window is a hassle that many homework helpers just won't bother with.

16. Aug 17, 2015

### DevonZA

You can view it easily on a computer. The writing is pretty clear and I am satisfied with the answer, I simply put it here as information.

17. Aug 17, 2015

### Ray Vickson

You should have left out the first sentence above; the correct statement replacing it should be something like "I am sorry, and won't do that in future".

I suggest you read the "pinned" post "Guidelines for students and helpers", by Vela (at the start of the list of messages). Many helpers will not bother to read attachments, even on a computer, so will not-- freely and without pay-- offer their time to help.

18. Aug 17, 2015

### DevonZA

I will type the answer out when I get some spare time. Thanks.

19. Aug 17, 2015

### Staff: Mentor

No need to do that for this thread, but keep it in mind for future posts.

20. Aug 17, 2015

### DevonZA

If there is no need to do it in this thread then why is Ray going on about it as if I have committed a crime? I did say that it is only there for information. I like to type my responses/answers/questions in correct latex form but for me to do this when I am satisfied with the answer and not actually asking anything seems a little extreme. Anyway I apologize if I upset anyone, I posted my final answer in PDF not knowing that it isn't a preferred method so it was unintentional.