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Oscillations: Pendulum with initial velocity 
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#1
Jan1613, 09:01 PM

P: 38

1. The problem statement, all variables and given/known data
A child on a 4m long swing is pulled back 1m from the vertical and released with a push imparting a speed of 2m/s. Find an expression for the angle θ, as a function of time, identifying the frequency ω, amplitude θi, and phase constant δ. 2. Relevant equations θ = θmax*cos(ωt + δ) ω = √g/l 3. The attempt at a solution I got a value for omega, from √g/l, of 1.565 The fact that the child starts with initial velocity is really throwing me off. In order to find the phase constant, I've tried looking at equations v = ωθ*sin(δ) θ = θi*cos(δ) and solving for tan(δ) to get a value for δ of 3.17rad... I don't even know if this is valid, but I otherwise don't know where to start. I know that if the child is starting from an angle of 0.253rad with initial velocity, he should travel passed the point where I can use small angle approximation. I thought also about differentiating the basic equation for harmonic motion θ = θmax*cos(ωt + δ) to get velocity, and setting that velocity equal to 2 at time 0, but there are still two unknowns. Any help will be appreciated. Thanks! 


#2
Jan1613, 11:36 PM

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θ(t)=Acos(ωt+δ) If Ω is the angular velocity of the pendulum, Ω=V/L Ω(t)=dθ/dt=Aωsin(ωt+δ) You push the child inward, so the direction of the initial velocity V is opposite to the displacement theta. Taking V negative you get a phase constant less than pi/2. The If you can apply small angle approximation depends on the maximum angular displacement, the amplitude. Get A by from the equation A^{2}=θ(0)^{2}+(Ω(0)/ω)^{2}. ehild 


#3
Jan1713, 12:01 AM

P: 38

I'm not sure I completely understand..
I am using θi to signify amplitude for a simple pendulum in the equations that you mentioned are incorrect, but when I look at how you say to find amplitude, θ(0) and dθ/dt (0) give me the equations you say are incorrect. θ(t)=Acos(ωt+δ) say I am using this, where A is amplitude instead of θi, if t = 0, θ(t)=Acos(δ) and if t = 0 here, Ω(t)=dθ/dt=Aωsin(ωt+δ) then dθ/dt=Aωsin(δ) And from those, would I not still need a phase shift value to solve the equation for A in A^2= A^2cos(δ)^2  A^2sin(δ)^2 ? Or am I just missing something blindingly obvious...? 


#4
Jan1713, 12:31 AM

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Oscillations: Pendulum with initial velocity
Note that the initial θ is not the amplitude. θ(0) = 0.253 rad, as you have found already and dθ/dt is initially 2(m/s)/4(m). ehild 


#5
Jan1713, 07:22 AM

P: 38

Okay, I'm starting to see it...
A^2= A^2cos(δ)^2 +A^2sin(δ)^2 where Acos(δ) = 0.253 = θ(0) and Aω*sin(δ) = 0.5 = dθ/dt at 0 so A^2 = (0.253)^2  0.5^2 and solve for A. I don't see how I would find the phase shift however. Does it involve dividing the sin and cos functions and solving for tan (δ)? 


#6
Jan1713, 08:27 AM

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ehild 


#7
Jan1713, 10:48 AM

P: 38

I have to divide the second term by ω = 1.565, yes?
A^2 = (0.253)^2  (0.5/1.565)^2 Then tan (δ) = AΩ(0)/Aωθ(0) I don't know how to determine which quadrant the function would be in, as in how to determine how sin and cos are positive or negative. Would only cos be negative if the denominator is negative in the equation? 


#8
Jan1713, 02:36 PM

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What is Ω(0) = V/L=2/4. So both sin(δ) and cos(δ) is positive, δ is in the first quadrant. In the second quadrant, sine is + and cosine is  In the third quadrant, bot sine and cosine are  In the fourth quadrant, sine is  and cosine is + ehild 


#9
Jan1713, 02:58 PM

P: 38

Ahh, forgot the fact that it is squared. Sorry that took so long to catch. I see it is positive now, not negative.
So ultimately, we have an equation where A = 0.408 tan(δ) = 0.204/0.162 = 0.33 in quadrant 1, δ = 0.033 so θ(t) = 0.408*cos(1.565t + 0.033) ? 


#10
Jan1713, 11:32 PM

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The amplitude is OK. I do not understand how you calculated delta. It is not right. Acosδ=0.253 Asinδ=0.319 Where did 0.204 and 0.162 come from? ehild 


#11
Jan1813, 02:26 AM

P: 38

tan (δ) = AΩ(0)/Aωθ(0)
I multiplied A by Ω(0) 0.408*(0.5) and A by ωθ(0) 0.408*1.565*2.53 = 0.162 I see now I already had the values for Acos(δ) and Asin(δ) tan(δ) = 0.319/0.253 δ = 0.9 so my equation for tan, tan (δ) = AΩ(0)/Aωθ(0) is actually incorrect, because I am changing the ratios if i choose to multiply by reciprocal... I needed to keep the divisions separate to keep the sine and cos ratios. silly silly mistake. sorry for all the trouble. should be tan (δ) = (Ω(0)/Aω)/((θ(0)/A) 


#12
Jan1813, 02:40 AM

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It is correct now. No need to divide by A, it cancels when you calculate the tangent.
ehild 


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