- #1
Physics Dad
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Homework Statement
Hi, I have the following question...
A pendulum consisting of a mass of 1kg is suspended from a string of length L. Air resistance causes a damping force of bv where b = 10-3 N/m
1) Derive and solve the equation of motion
2) Calculate the fractional decrease in amplitude of the pendulum oscillations if the pendulum is operated for 2 hours.
Homework Equations
F=ma
F=-kx
x(t)=A0eαt
The Attempt at a Solution
I think I have the derivation handled...
F=ma=-kx-bv
ma+bv+kx=0
m(d2x/dt2)+b(dx/dt)+kx=0
d2x/dt2+(b/m)(dx/dt)+(k/m)x=0
As this is a pendulum, I know x=Lsinθ and for small θ sinθ≈θ so...
d2θ/dt2+(b/m)(dθ/dt)+(k/m)θ=0
I also know that k=mg/L so...
d2θ/dt2+(b/m)(dθ/dt)+(g/L)θ=0
Therefore, I get the equation of motion for the dampened pendulum to be:
d2θ/dt2+(b/m)(dθ/dt)+(g/L)θ=0
Now to solve the equation, I can turn it into a quadratic:
α=(-(b/m)±(√(b2/m2)-(4g/L))/2
I can tidy this up a little, so...
α=-(b/2m)±(√(b2/2m2)-(4g/L))
I know that √g/L = ω so and -√g/L = iω which I can take out as a factor so and tidy up the fraction inside the radical, so...
α=-(b/2m)±iω√1-(b2L/2)
I also know that I can let ϑ=ω√1-(b2/4m2ω2) so...
α=-(b/2m)±iϑ
I am assuming that this is the equation solved as I don't have a value for L and so can't go any further (can I?)
As for calculating the fractional decrease, if I sub this back into the equation:
x(t)=A0eαt I get...
x(t)=A0e(-b/2m)t+eiϑt
But I really don't know where to go from here, if, if anywhere?
Do I simply rearrange the first part so...
x(t)/A0=e-18/5 so...
x(t)/A0=0.0273
Any help gratefully received!