(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If you guys cannot read the problem due to all the latex not pasting properly I will post the link to the website.

http://courses.ncsu.edu/py411/lec/001/; [Broken] then proceed to go to the homework section of my webpage then click on Assignment 6 and the problem is next to I.

PY411 — HOMEWORK #6

due October 13th

Read: Chapter 6. Problems: 5.5, 5.6, 5.9

I. The playground swing is a bit of a mystery! Simply by swinging your

feet at the right time (without directly experiencing an external force), the

amplitude of your swinging oscillations can grow with time. This is an example

of a parametric oscillator. To understand this phenomenon, we start

with an undamped simple penduluum described by the equation:

d2θ(t)

dt2 + ω2(t)θ(t) = 0,

where ω2 = ω2o

(1 + f (t)) and ωo =

pg

L ,with g the acceleration of gravity on

earth’s surface, L the penduluum length and f (t) is an oscillating function of

time (f (t) << 1). You can think of the oscillations as being due to periodic

changes in the length of the penduluum, which you cause by kicking your

feet. In order to keep things straightforward, pick a general solution for θ(t)

of:

θ(t) = A(t)eiωp t + B(t)e−iωp t,

and an oscillating term

f (t) =

fo

2i

(ei2ωp t − e−i2ωpt),

where ωp is the the angular frequency of the solution we are seeking. These

solutions and driving terms are not the most general we can select, but the

basic physics is included in this treatment.

i) As we did in class, plug in the general solution and collect terms with

the same time dependence, to derive the following equations which should

be true for all times:

d2A(t)

dt2 + i2ωp

dA(t)

dt

+

¡

ω2o

− ω2

p

¢

A(t) + ω2o

fo

2i

B(t) = 0,

d2B(t)

dt2 − i2ωp

dB(t)

dt

+

¡

ω2o

− ω2

p

¢

B(t) − ω2o

fo

2i

A(t) = 0.

1

Here we are making the important approximation that we can ignore “rapidly

oscillating terms” with oscillation frequencies of 3ωp. These terms do not

produce a cumulative effect (unlike the terms oscillating in phase with our

solutions proportional to eiωp t and e−iωpt, which do create such effects).

ii) Now we will seek solutions to the above equations where A(t) and B(t)

are written:

A(t) = r(t) cos αo,

B(t) = r(t) sinαo.

Once again, these are not the most general solutions (α is, in general, a

function of time, but the angle ultimately settles down to an equilibrium

value we can call αo). Plug these solutions into the equations above and

eliminate the terms proportional to d2r(t)

dt2 to derive the time dependence of

r(t).

iii) Can you actually start swinging fom a point where you are motionless

and your angular deflection θ(0) = 0? How does this differ from the standard,

driven oscillator?

iv) Where does the energy come from to increase the amplitude of your

swinging motion?

2. Relevant equations

3. The attempt at a solution

Isn't this a driven harmonic system problem since the child is making the swing move back and forth?

i) I think my general solution is [tex]\theta[/tex](t)= A(t)e^{iw(p)t}+Be^{-iw(p)t}

do I need to take the derrivative of theta twice to arrive at :

d^2A(t)/dt^2+iw(p)dA(t)/dt+(w(0)^2-w(p)^2)A(t)+w(0)^2*f(0)/2iB(t)=0

d^2B(t)/dt^2-iw(p)dA(t)/dt+(w(0)^2-w(p)^2)A(t)+w(0)^2*f(0)/2iB(t)=0

since f(t)<<1 should I just assume f(t) is negliglbe and w^2=g/L?

ii) For this part, I am just plugging in the first and second derivatives of A(t) and B(t) into the solutions I am supposed to derived in ii right?

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# Homework Help: Swing problem harmonic motion problem

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