Swing problem harmonic motion problem

[Benzoate]

Homework Statement

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/; 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?

Homework Equations

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 \theta(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?

 

[CompuChip]

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

I'd prefer to look at it as a "free" pendulum, but with a time-dependent length. I suppose you could also look at it as a very special case of a driven oscillator.

i) I think my general solution is \theta(t)= A(t)e^iw(p)t+Be^-iw(p)t

do I need to take the derrivative of theta twice

Yes

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

Excuse me? How do you get two expressions when deriving a single expression? It looks like you say "I take the second derivative twice" and then write down what you think you should be proving. Unfortunately, your answer will consist of more than one step Don't be impatient. What is the first derivative?

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

No, f is given, and if you look carefully, you see that f_0/2i is part of the final answer. You just plug it into your equation as well.

So, start by doing the derivations properly