An object of mass m=80 kg moves in one dimension subject to the potential energy
U(x)=λ4(x2−a2)2+b2x2.(1)
Here we use λ= 9 kg/(m2s2), a= 9 m, and b= 719 kg/s2.
(a) How many equilibrium points (stable and unstable ones) does this potential have?
n=
1
2
3
4
5?
(b) Find a stable equilibrium point x0 such that x0 is positive. (in meters)
x0=
(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)?
hint: For help on using Taylor series to express the potential energy function near a stable minimum, we encourage you to look at the following page in the section on small oscillations:here
F0=
k=
(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)
T=