Solving Simple Harmonic Motion: Find Angular Freq & Amp.

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Question: A particle undergoing simple harmonic motion has a velocity v1 when the displacement is x1 and a velocity v2 when the displacement is x2. Find the angular frequency and the amplitutde of the motion in terms of the given quantities. The answers given in the back of the book are: angular frequency = [(v2^2 - v1^2)/(x1^2 - x2^2)]^(1/2) and amplitude = [(x1^2*v2^2 - x2^2*v1^2)/(v2^2 - v1^2)]^(1/2).

I've tried approaching this several ways, but can't seem to get my answers to agree. Should I treat the solution as a sum of two separate solutions with two different amplittudes since this is a linear differential equation, such that x(t) = x1(t) + x2(t) = A*cos(w*t + phi) + B*cos(w*t + phi) and similarly with v(t) = v1(t) + v2(t) = -w*A*sin(w*t + phi) - w*B*sin(w*t + phi). Or is there an easier way? I'm not sure what is appropriate to generalize and what is not. For example, I've taken the phase angle phi to be zero which I believe is acceptable. Any other suggestions? Thanks.

 

[Tide]

You only need two arbitrary constants in your general solution. You can use either

[tex]A \cos(\omega t + \phi)[/tex]
or
[tex]A \cos \omega t + B \sin \omega t[/tex].

 

[quicknote]

I would first suggest reviewing the basic principles of simple harmonic motion and the equations involved. The angular frequency (ω) and amplitude (A) of a simple harmonic motion can be calculated using the equations ω = √(k/m) and A = x_max, where k is the spring constant and m is the mass of the particle. These equations hold true for any simple harmonic motion, regardless of the initial conditions.

In this case, we are given two different velocities (v1 and v2) and two corresponding displacements (x1 and x2). We can use these values to solve for the spring constant (k) and the mass (m) using the equations v1 = ω*A*cos(ω*t) and v2 = -ω*A*sin(ω*t). By setting these equations equal to the given velocities and solving for ω and A, we can find the angular frequency and amplitude of the motion.

Alternatively, we can use the given equations for angular frequency and amplitude in terms of the given quantities. However, it is important to note that these equations are derived from the general equations for simple harmonic motion and may not hold true for all cases. It would be best to check the derivation of these equations or consult a more reliable source to confirm their accuracy.

In terms of treating the solution as a sum of two separate solutions, this may not be necessary as the given information already accounts for the two different velocities and displacements. It would be more appropriate to use the equations for angular frequency and amplitude in terms of the given quantities.

In summary, it is important to review the basic principles and equations of simple harmonic motion, and carefully consider the given information before attempting to solve the problem. If there are any discrepancies or uncertainties, it is always best to seek guidance from reliable sources or consult with a colleague or mentor.