PGRE question: angular freq of small oscillations

[Aziza]

Problem from past PGRE:

A particle of mass m moves in a one-dimensional potential V(x)=-ax² + bx⁴, where a and b are positive constants. The angular frequency of small oscillations about the minima of the potential is equal to:

Answer is 2(a/m)^1/2.

I understand how this is found 'the long way' by actually applying calculus, but if you plug in a=-1/2 k and b=0 shouldn't you get back sqrt(k/m) ? Because the harmonic potential is 1/2kx^2 ... so where is the flaw in my logic?Thanks!

 

[ehild]

Problem from past PGRE:

A particle of mass m moves in a one-dimensional potential V(x)=-ax² + bx⁴, where a and b are positive constants. The angular frequency of small oscillations about the minima of the potential is equal to:

Answer is 2(a/m)^1/2.

I understand how this is found 'the long way' by actually applying calculus, but if you plug in a=-1/2 k, shouldn't you get back sqrt(k/m) ? Because the harmonic potential is 1/2kx^2 ... so where is the flaw in my logic?


Thanks!

If the potential was 1/2 kx²+bx⁴, there would be oscillation about x=0. But the problem says that both a and b are positive constants. So a can not be equal to -1/2 k where k is positive.

In case a>0 and b>0, the potential function has maximum at x=0. You do not get oscillatory motion about x=0. The minima are at x=±√(a/2b). Oscillatory motion is possible about a minimum of the potential function.