Young's modulus and uncertainty

[Jahnavi]

Homework Statement

Homework Equations

The Attempt at a Solution

Y = (F/A)/(∆L/L) = FL/A∆L

Putting values of F , L , A and simplifying

Y = 2×10⁹/∆L

I don't know how to proceed . I also don't understand what is uncertainty in length and uncertainty in strain .
Is uncertainty the same thing as error in the measurement of a quantity ?

Please help me with the problem .

 

[mfb]

Y = 2×10⁹/∆L

This is missing units.

Is uncertainty the same thing as error in the measurement of a quantity ?

Error is (usually) the colloquial name for uncertainty.

You are given the step size of length measurements. What is the Young's modulus that corresponds to a change by this step size under the given load?

 

[Jahnavi]

This is missing units

Sorry !

Y = 2×10⁹/∆L N/m²

You are given the step size of length measurements.

But why should ∆L be equal to least count of measurement of length i.e why should ∆L = .01mm ?

 

[mfb]

But why should ∆L be equal to least count of measurement of length i.e why should ∆L = .01mm ?

That is the smallest length change you can measure. It is interesting to see at which point the actual length change is equal to this smallest measurable change.

 

[Jahnavi]

That is the smallest length change you can measure.

∆L depends on the force applied and the properties of the material .

Least count is the smallest length measured by the instrument .

These are two separate things .

I still do not understand why should ∆L be equal to the least count ?

 

[mfb]

I still do not understand why should ∆L be equal to the least count ?

It does not have to be.
The question is "what happens if that is the case?" What can we say about a material that has this specific case?

 

[Jahnavi]

Is it right that since the least possible value of ∆L that can be measured will be equal to the least count , that smallest change will give the maximum possible value of Y that can be measured ?

In that case Y = 2 × 10¹⁴ N/m² which makes 3) false .

 

[mfb]

Is it right that since the least possible value of ∆L that can be measured will be equal to the least count , that smallest change will give the maximum possible value of Y that can be measured ?

Right.

In that case Y = 2 × 10¹⁴ N/m² which makes 3) false .

Good, that is one part done.

 

[Jahnavi]

Thanks .

Now I would like to check options 1) and 4) .

I will write Young's modulus as Y = FL/(πr²l)

L = length of the rod
l = Increase in the length of the rod
r = radius of cross section of the rod

∆Y/Y = ∆L/L + 2∆r/r + ∆l/l

Doubt 1) : Since least count is same for measuring lengths , does this mean ∆L = ∆l = ∆r ?

Doubt 2) : Minimum contribution in ∆Y/Y would be from ∆L/L which makes 1) option correct .

Is that correct ?

Doubt 3) :Contribution from error in strain would be the sum of 1st and 3rd terms i.e ∆L/L + ∆l/l .

But why would this contribution be maximum ?

Why wouldn't contribution of "error in radius " 2∆r/r i.e the 2nd term in ∆Y/Y be maximum ?

 

[mfb]

Since least count is same for measuring lengths , does this mean ∆L = ∆l = ∆r ?

I would expect that the old and new length have 0.01 mm uncertainty each, not the strain itself.

Minimum contribution in ∆Y/Y would be from ∆L/L which makes 1) option correct .

I agree.

Contribution from error in strain would be the sum of 1st and 3rd terms i.e ∆L/L + ∆l/l .

But why would this contribution be maximum ?

How large is 2∆r/r? How does this compare to typical ∆l/l (if you don't happen to measure a rubber band)?

 

[Jahnavi]

How large is 2∆r/r? How does this compare to typical ∆l/l

OK . Since r > l and ∆r = ∆l =.01mm , ∆r/r would be much smaller than ∆l/l .

Right ?

Is ∆l (change in increase in length ) comparable to l ( increase in length ) i.e ∆l/l close to being 1 ?

 

[mfb]

OK . Since r > l and ∆r = ∆l =.01mm , ∆r/r would be much smaller than ∆l/l .

Right ?

Right.

Is ∆l (change in increase in length ) comparable to l ( increase in length ) i.e ∆l/l close to being 1 ?

That will depend on your material.

 

[Jahnavi]

OK.

That leaves us with option 2) . What is a figure of merit ? What does it specify ?

 

[mfb]

That depends on the specific field, here I guess it is the inverse of the relative uncertainty.

 

[Jahnavi]

That depends on the specific field, here I guess it is the inverse of the relative uncertainty.

Is the figure of merit comparing L/∆L , r/∆r , l/∆l ?

Of the three ratios , L/∆L is the largest , making option 2) correct .

Right ?

Why is figure of merit defined in the way you have stated ( i.e inverse of relative uncertainty ) ?

 

[mfb]

Is the figure of merit comparing L/∆L , r/∆r , l/∆l ?

Of the three ratios , L/∆L is the largest , making option 2) correct .

Right ?

That's how I would interpret it.

Why is figure of merit defined in the way you have stated ( i.e inverse of relative uncertainty ) ?

That would be a typical definition, but in general such a figure of merit should be defined explicitly somewhere.

 

[Jahnavi]

Thanks !