Stationary Points [Challenging!] Differentiation: U = U_0 ((a/r)^(12)−2(a/r)^6)

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InfernoBear

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The potential energy of two molecules separated by a distance r is given by

[FONT=STIXMathJax_Main-italic]U [/FONT][FONT=STIXMathJax_Main]= [/FONT][FONT=STIXMathJax_Main-italic]U[/FONT][FONT=STIXMathJax_Main]0 [/FONT][FONT=STIXMathJax_Size1]([/FONT][FONT=STIXMathJax_Main]([/FONT][FONT=STIXMathJax_Main-italic]a[/FONT][FONT=STIXMathJax_Main]/[/FONT][FONT=STIXMathJax_Main-italic]r[/FONT][FONT=STIXMathJax_Main])[/FONT][FONT=STIXMathJax_Main]12[/FONT][FONT=STIXMathJax_Main]−[/FONT][FONT=STIXMathJax_Main]2[/FONT][FONT=STIXMathJax_Main]([/FONT][FONT=STIXMathJax_Main-italic]a[/FONT][FONT=STIXMathJax_Main]/[/FONT][FONT=STIXMathJax_Main-italic]r[/FONT][FONT=STIXMathJax_Main])[/FONT][FONT=STIXMathJax_Main]6[/FONT][FONT=STIXMathJax_Size1])[/FONT]


where [FONT=STIXMathJax_Main-italic]U[/FONT][FONT=STIXMathJax_Main]0[/FONT] and a are positive constants. The equilibrium separation of the two molecules occurs when the potential energy is a minimum; find the equilibrium separation and give the value of the potential energy at this separation if

[FONT=STIXMathJax_Main-italic]U[/FONT][FONT=STIXMathJax_Main]0 [/FONT][FONT=STIXMathJax_Main]= [/FONT][FONT=STIXMathJax_Main]6.62[/FONT][FONT=STIXMathJax_Main]×[/FONT][FONT=STIXMathJax_Main]10[/FONT][FONT=STIXMathJax_Main]−[/FONT][FONT=STIXMathJax_Main]21[/FONT][FONT=STIXMathJax_Main-italic]J
[/FONT]
This was set by a very tough teacher - and we have only just started simple differentiation (as in x^3 = 3x^2) I dont know how to even start this problem
 
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InfernoBear said:
The potential energy of two molecules separated by a distance [FONT=STIXMathJax_Main-italic]rr is given by [FONT=STIXMathJax_Main-italic]U[/FONT][FONT=STIXMathJax_Main]=[/FONT][FONT=STIXMathJax_Main-italic]U[/FONT][FONT=STIXMathJax_Main]0[/FONT][FONT=STIXMathJax_Size1]([/FONT][FONT=STIXMathJax_Main]([/FONT][FONT=STIXMathJax_Main-italic]a[/FONT][FONT=STIXMathJax_Main]/[/FONT][FONT=STIXMathJax_Main-italic]r[/FONT][FONT=STIXMathJax_Main])[/FONT][FONT=STIXMathJax_Main]12[/FONT][FONT=STIXMathJax_Main]−[/FONT][FONT=STIXMathJax_Main]2[/FONT][FONT=STIXMathJax_Main]([/FONT][FONT=STIXMathJax_Main-italic]a[/FONT][FONT=STIXMathJax_Main]/[/FONT][FONT=STIXMathJax_Main-italic]r[/FONT][FONT=STIXMathJax_Main])[/FONT][FONT=STIXMathJax_Main]6[/FONT][FONT=STIXMathJax_Size1])[/FONT]U=U0((a/r)12−2(a/r)6) where [FONT=STIXMathJax_Main-italic]U[/FONT][FONT=STIXMathJax_Main]0[/FONT]U0 and [FONT=STIXMathJax_Main-italic]a[/FONT]a are positive constants. The equilibrium separation of the two molecules occurs when the potential energy is a minimum; find the equilibrium separation and give the value of the potential energy at this separation if [FONT=STIXMathJax_Main-italic]U[/FONT][FONT=STIXMathJax_Main]0[/FONT][FONT=STIXMathJax_Main]=[/FONT][FONT=STIXMathJax_Main]6.62[/FONT][FONT=STIXMathJax_Main]×[/FONT][FONT=STIXMathJax_Main]10[/FONT][FONT=STIXMathJax_Main]−[/FONT][FONT=STIXMathJax_Main]21[/FONT][FONT=STIXMathJax_Main-italic]J[/FONT]U0=6.62×10−21J.[/FONT]
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InfernoBear said:
The potential energy of two molecules separated by a distance r is given by

[FONT=STIXMathJax_Main-italic]U [/FONT][FONT=STIXMathJax_Main]= [/FONT][FONT=STIXMathJax_Main-italic]U[/FONT][FONT=STIXMathJax_Main]0 [/FONT][FONT=STIXMathJax_Size1]([/FONT][FONT=STIXMathJax_Main]([/FONT][FONT=STIXMathJax_Main-italic]a[/FONT][FONT=STIXMathJax_Main]/[/FONT][FONT=STIXMathJax_Main-italic]r[/FONT][FONT=STIXMathJax_Main])[/FONT][FONT=STIXMathJax_Main]12[/FONT][FONT=STIXMathJax_Main]−[/FONT][FONT=STIXMathJax_Main]2[/FONT][FONT=STIXMathJax_Main]([/FONT][FONT=STIXMathJax_Main-italic]a[/FONT][FONT=STIXMathJax_Main]/[/FONT][FONT=STIXMathJax_Main-italic]r[/FONT][FONT=STIXMathJax_Main])[/FONT][FONT=STIXMathJax_Main]6[/FONT][FONT=STIXMathJax_Size1])[/FONT]


where [FONT=STIXMathJax_Main-italic]U[/FONT][FONT=STIXMathJax_Main]0[/FONT] and a are positive constants. The equilibrium separation of the two molecules occurs when the potential energy is a minimum; find the equilibrium separation and give the value of the potential energy at this separation if

[FONT=STIXMathJax_Main-italic]U[/FONT][FONT=STIXMathJax_Main]0 [/FONT][FONT=STIXMathJax_Main]= [/FONT][FONT=STIXMathJax_Main]6.62[/FONT][FONT=STIXMathJax_Main]×[/FONT][FONT=STIXMathJax_Main]10[/FONT][FONT=STIXMathJax_Main]−[/FONT][FONT=STIXMathJax_Main]21[/FONT][FONT=STIXMathJax_Main-italic]J
[/FONT]
This was set by a very tough teacher - and we have only just started simple differentiation (as in x^3 = 3x^2) I dont know how to even start this problem
Click to expand...

\(\displaystyle \dfrac{d}{dr} \ = \ 0\) for local minima or maxima.

You are saying that this was not discussed in the Physics or Math class - prior to this assignment!!!
 
Subhotosh Khan said:
\(\displaystyle \dfrac{d}{dr} \ = \ 0\) for local minima or maxima.

You are saying that this was not discussed in the Physics or Math class - prior to this assignment!!!
Click to expand...


Yes!!!!!
We have the toughest teachers for physics in the school he doesn't tell us how to do the homework he just sets it!!!!!
 
InfernoBear said:
Yes!!!!!
We have the toughest teachers for physics in the school he doesn't tell us how to do the homework he just sets it!!!!!
Click to expand...

Then - how did you know that you are supposed to differentiate the function!!
 
InfernoBear said:
Thats all we were taught in the lesson
Click to expand...
Okay; so you've been taught the differentiation rule for pn, where n is an integer and p is the variable. In this exercise, you've been given that U is the variable, and you've been told to find the minimum (so find the derivative and set it equal to zero) for terms of the form Un.

Try applying the derivative rule they gave you, and then following the "set it equal to zero and solve" process they gave you. Where does this lead? ;)
 
Since you know how to differentiate "\(\displaystyle x^n\)", the crucial point in this problem is to recognize that powers in the denominator can be written as negative powers in the numerator. That is, \(\displaystyle U = U_0 ((a/r)^{12}−2(a/r)^6)= U_0(a^{12}r^{-12}- 2a^6r^{-6})\).

Now, since the derivative of \(\displaystyle r^n\), with respect to r, is \(\displaystyle nr^{n-1}\), what are the derivatives of \(\displaystyle r^{-12}\) and \(\displaystyle r^{-6}\)?
 
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