- #1
JC2000
- 186
- 16
Elastic Potential Energy of a Strained Body
(A) Using ## Y = \frac {stress}{strain}## we get ##F = \frac {AY}{L} * x## where ##F## is the restoring force, ##x## is the distance the body is stretched by.
Since Work = PE (spring force/ stress is conservative?)
Thus ##W = \int_{0}^{x} \frac {AY}{L} x dx ## which gives ##W = \frac {AY}{2L} x^2## (?)
OR
(B) For spring force we know ##F = kx## thus ## W = \frac {kx^2}{2}##
Comparing ##F=kx## to ##Y = \frac{FL}{Ax}## gives us ##k=\frac{FA}{L}## substituting this in the above result gives us ##W = \frac {AY}{2L} x^2##
My Questions :
1. Are both derivations correct?
2. My book also mentions ##U = 1/2 * Stress * Strain * Volume##, I am (a)not sure if volume refers to ##x^3## or ##L^3## and (b) I am unable to derive this result from ##W = \frac {AY}{2L} x^2## (apologies for the trivial question).
Slightly related :
3. I was solving a problem (A wire of mass ##m## and length ##l## is suspended from the ceiling. Due to its own weight it elongates, consider cross-section area ##A## and Young's modulus ##Y##. Find the elongation of the wire.) which also involves integration(?).
The solution :
Assume a small length of the wire ##dx## which elongates by ##\Delta dx## so that ##Y = \frac{T/A}{\Delta dx/dx}## which can be expressed as :
## \Delta dx = \frac {T}{YA}dx = \frac{mg}{YAl}*xdx##.
Thus, total elongation ##\Delta l = \int \Delta dx = \int_{0}^{l} \frac {mg}{YAl}*xdx = \frac{mgl}{2YA}##
Here I have two questions (c) Why is the fact that each small length of wire considered undergoes a slightly different F, ignored (I can't fathom how to deal with this though, some sort of 'double integration' possibly?)?
(d) How does ##\frac {T}{YA}dx = \frac{mg}{YAl}*xdx##
4. Lastly, throughout the derivation for stretching the body, signs are ignored, If I understand correctly. signs can be assigned as per convenience (for compression/ stretching or for work done by restoring force/ work done by system) and the chosen signs need to be mentioned initially for the sake of consistency (?). Is this rigorous enough? Or is there some convention that is generally followed here?
Thank you for your time!
(A) Using ## Y = \frac {stress}{strain}## we get ##F = \frac {AY}{L} * x## where ##F## is the restoring force, ##x## is the distance the body is stretched by.
Since Work = PE (spring force/ stress is conservative?)
Thus ##W = \int_{0}^{x} \frac {AY}{L} x dx ## which gives ##W = \frac {AY}{2L} x^2## (?)
OR
(B) For spring force we know ##F = kx## thus ## W = \frac {kx^2}{2}##
Comparing ##F=kx## to ##Y = \frac{FL}{Ax}## gives us ##k=\frac{FA}{L}## substituting this in the above result gives us ##W = \frac {AY}{2L} x^2##
My Questions :
1. Are both derivations correct?
2. My book also mentions ##U = 1/2 * Stress * Strain * Volume##, I am (a)not sure if volume refers to ##x^3## or ##L^3## and (b) I am unable to derive this result from ##W = \frac {AY}{2L} x^2## (apologies for the trivial question).
Slightly related :
3. I was solving a problem (A wire of mass ##m## and length ##l## is suspended from the ceiling. Due to its own weight it elongates, consider cross-section area ##A## and Young's modulus ##Y##. Find the elongation of the wire.) which also involves integration(?).
The solution :
Assume a small length of the wire ##dx## which elongates by ##\Delta dx## so that ##Y = \frac{T/A}{\Delta dx/dx}## which can be expressed as :
## \Delta dx = \frac {T}{YA}dx = \frac{mg}{YAl}*xdx##.
Thus, total elongation ##\Delta l = \int \Delta dx = \int_{0}^{l} \frac {mg}{YAl}*xdx = \frac{mgl}{2YA}##
Here I have two questions (c) Why is the fact that each small length of wire considered undergoes a slightly different F, ignored (I can't fathom how to deal with this though, some sort of 'double integration' possibly?)?
(d) How does ##\frac {T}{YA}dx = \frac{mg}{YAl}*xdx##
4. Lastly, throughout the derivation for stretching the body, signs are ignored, If I understand correctly. signs can be assigned as per convenience (for compression/ stretching or for work done by restoring force/ work done by system) and the chosen signs need to be mentioned initially for the sake of consistency (?). Is this rigorous enough? Or is there some convention that is generally followed here?
Thank you for your time!