Go Back   Physics Help Forum > College/University Physics Help > Advanced Mechanics

Advanced Mechanics Advanced Mechanics Physics Help Forum

Reply
 
LinkBack Thread Tools Display Modes
Old Mar 8th 2014, 08:58 AM   #1
Member
 
Join Date: Feb 2013
Location: Greater St. Louis area
Posts: 43
Angular frequency

A mass moves along the x axis with potential energy
U(x)= - U0 a^2 / (a^2 + x^2). What is the angular frequency assuming the oscillation is small enough to be harmonic?

w^2 = k/m with w as the angular frequency

F= -kx = -(gradient) U

Since this is one-dimensional we take the derivative of U with respect to x.

I get -(gradient) U = -2 U0 a^2 x / (a^2 + x^2)^2

Therefore k= 2 U0 a^2 / (a^2 + x^2)^2

The correct answer does not have an x term in it.

Is there a binomial expansion that would essentially eliminate the x term in the denominator?

Thanks for any help.
roger is offline   Reply With Quote
Old Mar 10th 2014, 06:46 AM   #2
Physics Team
 
ChipB's Avatar
 
Join Date: Jun 2010
Location: Morristown, NJ USA
Posts: 2,290
If the motion is harmonic then x is a function of time:

x = A sin(wt)

where w = rotational velocity in radians/second. The equation of motion from conservation of energy is:

m(d^2x/dt^2)+U = 0. [**EDIT - this is wrong - see correction in post #4 **]

So:

-mAw^2 sin(wt)+2Asin(wt)cos(wt)U_0a^2/(a^2+A^2sin^2(wt))^2 = 0

Can you take it from here? You can apply a couple of trig identities for cos(2x) and sin(2x) to simplify things.

Last edited by ChipB; Mar 11th 2014 at 06:26 AM.
ChipB is online now   Reply With Quote
Old Mar 10th 2014, 04:28 PM   #3
Member
 
Join Date: Feb 2013
Location: Greater St. Louis area
Posts: 43
Thank you

Thank you so much Chip.

Yes, I can solve this now.
roger is offline   Reply With Quote
Old Mar 11th 2014, 06:26 AM   #4
Physics Team
 
ChipB's Avatar
 
Join Date: Jun 2010
Location: Morristown, NJ USA
Posts: 2,290
Upon review, I'm afraid this part is incorrect:

Originally Posted by ChipB View Post
m(d^2x/dt^2)+U = 0.
This should be: m(d^2x/dt^2) + dU/dx = 0.

Sorry for this.
ChipB is online now   Reply With Quote
Reply

  Physics Help Forum > College/University Physics Help > Advanced Mechanics

Tags
angular, frequency



Thread Tools
Display Modes


Similar Physics Forum Discussions
Thread Thread Starter Forum Replies Last Post
Resonance,Resonance frequency,natural frequency? hyungjko Waves and Sound 1 Feb 7th 2013 12:33 PM
Angular Frequency Question help need sonog4 Periodic and Circular Motion 1 May 4th 2011 05:09 AM
Angular Frequency in Magnetic field SVXX Electricity and Magnetism 1 Jun 3rd 2009 07:05 AM
Classical Mechanics: angular frequency kala Advanced Mechanics 1 Dec 4th 2008 09:19 PM
Frequency werehk Waves and Sound 1 Aug 9th 2008 07:11 AM


Facebook Twitter Google+ RSS Feed