Rigid Bodies

Question 1

What is a good example to use for a rigid body?

Answer 1

Solid ball rotating about axis n(hat)(normal to the top) with angular velocity ω = ω n(hat)

Question 2

What are 2 important points to consider when dealing with rigid bodies?

Answer 2

Distance between the particles is fixed. Every particle has the same ω, but different ri, vi, where vi = ωxri

Question 3

What is the equation for the total kinetic energy of all the particles in the ball?

Answer 3

T = 1/2sum over i of mivi^2 = 1/2sum over i of miω^2*ri^2

Question 4

How can we define the moment of inertia?

Answer 4

Take ω out of the kinetic energy equation and the moment of inertia is the sum part, so I = sum over i of miri^2, making the KE T = 1/2ω^2*I

Question 5

What is an alternative equation for the moment of inertia I?

Answer 5

I = integral over body of r^2 dm = integral over body of r^2 ρ dV

Question 6

What is the constraint we can use for a gyroscope?

Answer 6

Centre of mass is at a fixed distance from the origin

Question 7

What coordinates do we use for the gyroscope problem?

Answer 7

B, Ф, θ, where B is the rotation, Ф is the angle around x-y plane and θ is the angle down from z-axis

Question 8

How many moments of inertia does the gyroscope have and what are they?

Answer 8

3 moments of inertia: B’ = B’ B(hat) (I around axis associated with B, I for rotation around z-axis (angular velocity Ф’ = Ф’ z(hat), and I for rotation involving θ’ which is the same for rotation around z because of the axial rotation symmetry.

Question 9

What is the simplest way to look at the angular velocities of the gyroscope?

Answer 9

Get 3 orthogonal angular velocities -> can break problem up into a form where we can get T easily.

Question 10

What is the first angular velocity we break the gyroscope up into?

Answer 10

Along axis of gyro: B’ + Ф’cosθ

Question 11

What is the second angular velocity we break the gyroscope up into?

Answer 11

x-y plane perpendicular to gyro axis: θ’

Question 12

What is the third angular velocity we break the gyroscope up into?

Answer 12

perpendicular to axis and θ’:Ф’sinθ

Question 13

How do we find the total kinetic energy over all rotation axis?

Answer 13

T = 1/2 * sum over i of Ii*ωi^2, where I is the moment of inertia and i is the rotation index

Question 14

What is the subbed-in equation for the KE T?

Answer 14

T = 1/2J(B’+ Ф’cosθ) + 1/2Iθ’^2 + 1/2I*Ф’^2 sin^2θ

Question 15

What is the equation for the potential energy V of the gyroscope?

Answer 15

V = mgl*cosθ, where l is the distance of centre of mass from the origin or a point.

Question 16

What do we do after we have the kinetic energy and the potential energy for the gyroscope?

Answer 16

Find the Lagrangian L = T-V

Question 17

How do we find the canonical momenta for each axis?

Answer 17

Differentiate L with respect to each variable: p(θ) = dL/dθ’, p(Ф) = dL/dФ’ etc

Question 18

What is the Lagrangian a function of? What are the conserved quantities for the gyroscope?

Answer 18

L(θ, θ’, Ф’, B’). p(Ф) and p(B) are conserved as Ф & B aren’t in L. L is not an explicit function of time so H is conserved.

Question 19

What is the equation for the Hamiltonian for the gyroscope?

Answer 19

H = T+V = E

Question 20

How do we write the Hamiltonian H in terms of the canonical momenta: p(θ), p(Ф) and p(B)?

Answer 20

Do T+V and then sub in th equations for the canonical momenta to get H(θ, p(θ), p(Ф), p(B)

Question 21

What is the Hamiltonian in terms of the canonical momenta?

Answer 21

H = p(θ)/2I + ((p(Ф)-p(B)cosθ)^2/(2Isin^2(θ)) + p(B)^2/2J + mgl*cosθ

Question 22

What is another way we can write the Hamiltonian in terms of the canonical momenta and why?

Answer 22

H = p(θ)/2I + Ueff(θ) = E, where Ueff(θ) is the effective potential energy.

Question 23

How do we get an equation for θ’ from the Hamiltonian?

Answer 23

Use p(θ) = I*θ’, and sub it into the Hamiltonian. Then rearrange for θ.