Approximations for small oscillations

In summary, the conversation discusses an issue with Landau & Lifgarbagez mechanics and the formula for δl. The person is unsure how to obtain the formula on the far right, which involves a Taylor expansion. Suggestions are given for manipulating the formula and using Taylor expansions for small angles in radians.
  • #1
Maybe_Memorie
353
0

Homework Statement



Basically the issue is Landau & Lifgarbagez mechanics says

δl = [r2 + (l + r)2 - 2r(l + r)cosθ]1/2 - l ≈ r(l + r)θ2/2l


Homework Equations



θ much less than 1

The Attempt at a Solution



I've no idea how to get the thing on the far right. I'm assuming it's Taylor expansion or something like that but that's not something any of my classes have really explained. Also this isn't homework, I'm on summer holidays.
 
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  • #2
Can you massage the formula into
[tex] \delta l = l \sqrt{1 + \frac{2r(l+r)}{l^2}(1-\cos \theta)} - l [/tex]? Then all you need to do is to do Taylor expansions for 1-cos θ and √(1+x).
 
  • #3
try using the cos(theta)= 1 - (theta^2)/2 approx for small angles in radians
 

1. What are small oscillations?

Small oscillations refer to the motion of a system around its equilibrium position, where the amplitude of the oscillation is small compared to the overall size of the system. This type of motion is governed by simple harmonic motion and is often seen in pendulums, springs, and other mechanical systems.

2. Why are approximations used for small oscillations?

Approximations are used for small oscillations because they simplify the mathematical analysis of the system. By assuming small oscillations, we can use linear approximations and ignore higher order terms in the equations of motion, making the calculations easier and more manageable.

3. What is the formula for calculating the period of small oscillations?

The period of small oscillations can be calculated using the formula T = 2π√(m/k), where T is the period, m is the mass of the oscillating object, and k is the spring constant of the system.

4. How can we determine the stability of a system with small oscillations?

The stability of a system with small oscillations can be determined by analyzing the eigenvalues of the system's matrix of coefficients. If all eigenvalues are negative, the system is stable. If any eigenvalue is positive, the system is unstable.

5. Can small oscillations occur in nonlinear systems?

Yes, small oscillations can occur in nonlinear systems. However, the equations of motion for these systems will not be simple harmonic motion and will require more complex analysis and techniques, such as perturbation methods, to approximate the solutions.

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