An object of mass m=80 kg moves in one dimension subject to the potential energy
U(x)=λ/4*(x2−a2)2+(b/2*x2)
a) How many equilibrium points (stable and unstable ones) does this potential have?
b) Find a stable equilibrium point x0 such that x0 is positive. (in meters)
c) Do a Taylor expansion of the force F(x) for x close to the equilibrium point, x≃x0, that is F(x)=F0−k(x−x0)+… What are the values for F0 (in Newton) and k (in kg/s2)?
(d) What is the period T of small oscillations (in seconds) of this mass around the equilibrium point x0? (Note that the parameter k found in the previous question acts like a spring constant that wants to pull small deviations back to the equilibrium point)
a)3
b)take the derivative of U(x).. U'(x)
then take the zero-points U'(x)=0.. one zero is x=0, and it has two zeros symmetric to the y-axis.. take the positive one as answer for x0.
c)wolframalpha will do this for you. enter taylor expansion take U'(x) as the function and your x0..
d)2*pi/sqrt(k/m) (k from c) )
@bw
could your please tell part c properly
I couldn't understand part c. In wolframalpha, what do we have to enter as the order of the taylor series and the point?
@phy
what other questions did you got?
See... for x=x0... if I do my taylor serie expansion... my k is being accepted as correct but my F0 is not being accepted by edx... wonder why...
@anokneemouse what is d formula for part d?
what value of k did you got?
my x_o is 7.74
a=7 and b =-11 lamda=1
ss01
look.... its quite simple.... F0... comes to 0 for mine... and to get k, do the following:
Since I don't know the values of a,b,lambda you have...
F(x)=dU(x)/dx
now type in F(x) in wolfram alpha... you'll get 3 roots...
1)x=0
and 2 more equal roots with opposite signs.
the positive root must be chosen.
Now in wolfram alpha type in Talyor Expansion( and type in your F(x)) at x=the positive root.
the second term's coefficient is k.
d) T=2*pi/sqrt(k/m)
:) Cheers...
Mouse/Anokneemouse.
PS - I need help with the airliner and mass spring and pendulum sums
what abt the rotating disc?
thnx mouse