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BlackAntlers
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Homework Statement
Hello everyone, first time poster. I am in an Analytical Mechanics class and we are covering Dimensionless Form and Free Fall and am struggling with some of the concepts. The question has multiple parts, but looks like this. The equation for a damped harmonic oscillator is,
[itex]m\ddot{x} = -kx + b\dot{x}[/itex]
a) What are the units of b?
b) What two variables will we need to make dimensionless by scaling?
c) For undamped harmonic motion what is the unit of time and how is it related to the spring constant K and the mass m?
d) Using this natural unit of time, construct the transformation that maps time to a dimensionless variable.
e) Using the time transformation you developed, turn the original equation into
[itex]
{\frac{d^2 x}{dt^*2}} = -x + α {\frac{dx}{dt^*}}
[/itex]
f) Identify α in terms of k, m and b. If you computed correctly, α should be dimensionless.
g)We have other constants available for scaling our diff eq. other than the ones that appear explicitly. We have two additional candidates for length and velocity, what are they and where do they come from?
Homework Equations
Period of a spring constant [itex] T = 2π\sqrt\frac{m}{k}
[/itex]
The Attempt at a Solution
a) We know mass times acceleration is a force F, whose units are N. N =[itex] \frac {mass*length}{time^2}[/itex]. We know velocity [itex]\dot{x} = \frac{length}{time}[/itex] so, b must be [itex]\frac{mass}{time}.[/itex]
b) Having an issue with this one because don't we want all variables that have dimensions to be dimensionless? Like mass, time, length and from those we could figure out acceleration and velocity in dimensionless terms?
c) The period for undamped harmonic motion is in seconds. The equation is in the relevant equations part. To figure out what K is, look in the original equation. X is a length, so K = [itex]\frac{mass}{seconds^2}.[/itex] When we cancel out our units in the period equation, we are left with seconds, which checks out. My only question is about the 2π. What do we do with this? Do we just ignore it?
d) Ok, so our [itex]T = \sqrt\frac{M}{K} * T^*, (T^*)[/itex] is our dimensionless unit for time.
Then [itex]dT = \sqrt\frac{m}{k}*dT^*[/itex] and [itex]dT^2 = \frac{m}{k}*T^{2*}[/itex]
e) Dividing the mass out in original equation [itex]\frac{d^2 x}{m/k}\frac{1}{dT^{2*}} = \frac{-k*x}{m} + \frac{b}{m} \frac{dx}{sqrt(m/k)}\frac{1}{dT^*}. [/itex] So, distributing the [itex]\frac{m}{k}[/itex] and cleaning up we get [itex]{\frac{d^2 x}{dt^*2}} = -x + α {\frac{dx}{dt^*}}[/itex]
f) I get [itex] α = \frac{b}{m} \frac{m}{k}[/itex] and when I plug in all the units they end up canceling out to be dimensionless, so that checks out.
g) This is the part I'm really stuck on. What are the two additional candidates for length and velocity? Do I have to add an [itex]x_o[/itex] and [itex]v_o[/itex] to the previous equation? And if I do, why exactly do I do this?
Thank you all in advance