Calculate 1st Order Correction to Hooke's Law Period

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The discussion focuses on calculating the first-order correction to the period of oscillation for a block attached to a non-ideal spring, described by the force equation F = -kx - Bx^3. The original period for an ideal spring is T = 2π√(m/k), but the presence of the cubic term suggests the period will be shorter. Participants explore methods to derive the new angular frequency, including using acceleration as a function of spring stretch and applying conservation of energy principles. One user suggests assuming a periodic solution and deriving the necessary equations, ultimately arriving at an expression for angular frequency that incorporates the correction term. The conversation emphasizes the need to carefully consider the effects of the non-ideal spring on the oscillation period.
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Homework Statement


a block of mass m rests on a horizontal frictionless surface. The block is attached to a horizontal spring.

The spring is not ideal. The force exerted on the block by the spring is F= -kx-Bx^3 where B is a positive constant.

Calculate the first-order (largest non-zero) correction to the hooke's law period when the amplitude of the motion of the block is small, but finite


Homework Equations



If it was ideal the period of the oscillation would be T= 2*pi * sqrt of (m/k)
but it is not idea

The Attempt at a Solution



I know the period has to be shorter because the restraining force is greater now with the added -Bx^3 term but the questions is how much shorter is the period?

I just tried solving for k=-F/x - Bx^2 and then plugging it back into the period formula but that doesn't seem right

Ok I know that -kx-Bx^3= m*x" (2nd deriv of x with respect to time)

so I thought maybe I could solve it like an ode

hmm
 
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You have a force equation as a function of spring stretch. Remember force relates to acceleration, so you really have acceleration as a function of spring stretch. To relate to period, you really want to find the speed (as a function of spring stretch). You can do that a few ways: via acceleration or via the "potential" surface and conservation of energy (the method I tend to prefer and suggest). It's typically at that point that you decide to call the speed x-dot and separate separate variables to find angular frequency, which can be related to period.

I suggest following the text case with the non-altered force equation to work through this process... then do again with the added force (you might be able to think of it as a unique driving force).
 
This is what I tried

I know its periodic so I simply assumed that solution is x=Acos(wt) then I took the first and second derivatives for x' and x"

then I simply plugged those back into the equation keeping in mind that I have cos^3 term for x^3

using a trig identity I solved for it and got w= sqrt of [(k +.75BA^2)/mass]

please tell me if I am way off the mark
 

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