Spinning Top Equations of Motion

  • #1
latentcorpse
1,444
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ok so I've got the three components of the inertia tensor (A,A,C) and I've derived expression for, using matrices, the angular velocities w1,w2,w3 (these expressions can be found online if you don't already know them). anywho, what i need to do next is use the fact that torque = rate of change of angular momentum to derive equations for the three components of torque.
so, for example,

tau 3 = C*d/dt{w3}
this cancels out quite nicely and we get that tau 3 =0.

However, when trying to get expressions for tau 1 and 2 the algebra gets very messy.

Help!

sorry if this isn't clear enough but if you don't understand what i mean just post and i'll get back to you!
 

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
25,838
256
Welcome to PF!

However, when trying to get expressions for tau 1 and 2 the algebra gets very messy.

Hi latentcorpse! Welcome to PF! :smile:

Show us the expressions you got, so that we can help! :smile:
 
  • #3
latentcorpse
1,444
0
okay so I've got (w is omega):
angular velocities:

w1= theta(dot)*sin(psi) - phi(dot)*sin(theta)cos(psi)
w2= theta(dot)*cos(psi) - phi(dot)*sin(theta)sin(psi)
w3= psi(dot) + phi(dot)*cos(theta)

Then the kinetic energy,T, is
T= 1/2*I*w^2=1/2*(A*w1^2+A*w2^2+C*w3^2)
this expression simplifies when you square and add

the potential energy is V=Mgh*cos(theta)

and so the lagrangian is

L=T-V = 1/2*A*(phi(dot))^2 + 1/2*A*(phi(dot))^2*(sin(theta))^2 +
1/2*C*[psi(dot)+phi(dot)*cos(theta)]^2 - Mgh*cos(theta)

anyway it's definitely right to this point.

so now we do :
(*)
d/dt[dL/d(psi(dot))] - dL/d(psi) = 0

but dL/d(psi) = 0

so,

d/dt[dL/d(psi(dot))] =
C[psi(doubledot) + phi(doubledot)*cos(theta) - phi(dot)*theta(dot)*sin(theta)] = 0
Noting that the term inside the square brackets in the above line is actually w3(dot)
we get:

C*w3(dot) = 0

But tau = I*w(dot) where I is the moment of inertia

and so tau3 = 0

Now we need to do a similar processfrom the (*) onwards but for theta and phi and hopefully get somewhere but I keep getting stuck but my tutor assures me it should work. argh!
also sorry for writing it out longhand i haven't learned lateX yet.
 

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