Rotating Reference Frames (Ch 9)
Give the formula for the Coriolis force.
Fcor = 2mr' × Ω
Write the matrix form of the equation of motion for small oscillations.
M q" = -K q
Write the integral formulas for the diagonal and off-diagonal elements of a moment of inertia tensor.
Ixx = ∫(y2 + z2)dm
Ixy = -∫(xy)dm
Write the Euler-Lagrange equation!
Give the formula for the centrifugal force.
Fcf = m(Ω × r) × Ω
Write the eigenvalue equation for normal modes.
(K - ω2M)a = 0
Write Euler's equations (relation between angular momentum, angular velocity, torque...)
L' + ω × L = Γ
(can write this in component form)
A particle is moving in a circular (counterclockwise) path confined to the xy plane. In what direction does the angular momentum vector point?
What are the directions of the centrifugal and Coriolis forces on a person moving (a) south near the North Pole, (b) east on the equator, and (c) south across the equator?
(Taylor 9.8)
(a) Fcf is south (and a little up), Fcor is west.
(b) Fcf is vertically up, Fcor is vertically up.
(c) Fcf is vertically up, Fcor is zero.
How many normal modes are there for 2 particles attached by a spring in 1D? What about 3 particles attached by springs in 1D? N particles?
2 particles => 2 normal modes
3 particles => 3 normal modes
N particles => N normal modes
The number of normal modes corresponds to the number of degrees of freedom in the system.
When is the inertia tensor diagonal? What are the values of the diagonal components in this case?
The inertia tensor is diagonal when evaluated with respect to the principle axes. In this case, the diagonal elements are equal to the principle moments
How do you find the eigenvalues and eigenvectors of a matrix?
Eigenvalues: solve det(A - λ1) = 0 for λ.
Eigenvectors: solve (A - λ1)v = 0 for v.
I wish to place a puck on a rotating horizontal turntable (angular velocity Ω) and to have it remain at rest on the table, held by the force of static friction (coefficient μ). What is the maximum distance from the axis of rotation at which I can do this? (Argue from the point of view of an observer in the rotating frame.)
(Taylor 9.12)
For the puck on the rotating horizontal turntable, there are four forces, its weight mg, the normal force N of the table, the force of friction f, and the centrifugal force. If the puck is not to move on the table these must sum to zero, mg+N+f+Fcf = 0. The two vertical forces must balance, so N = mg, and the two horizontal forces must also balance, so Fcf = mΩ2r = f ≤ μN = μmg. Therefore r ≤ μg/Ω2.
Find the normal frequencies for the system of two carts and three springs shown in Figure 11.1 (Taylor pg. 418), for arbitrary values of m1 and m2 and of k1, k2, and k3. Check that your answer is correct for the case that m1 = m2 and k1 = k2 = k3.
(Taylor 11.3)
ω2 = (1/2m1m2) {m1(k2 + k3) + m2(k1 + k2) ± √(m12 (k2 + k3)2 + m22 (k1 + k2)2 +2m1m2(k22 - k1k2 - k1k3 - k2k3) }
if m1=m2=m and k1=k2=k3=k, then ω2 = (k/m)(2±1)
Consider a lamina, such as a flat piece of sheet metal, rotating about a point O in the body. Prove that the axis through O and perpendicular to the plane is a principal axis.
(Taylor 10.30)
Lamina => Ixz=Izx=Iyz=Izy=0.
If ω points along the z axis, then ω has components (0,0,ω), and L= Iω = (0,0, Izzω) (both to be thought of as column vectors), which also points along the z axis. Therefore the z axis is a principal axis.
Verify that the components of the vector r × (ω × r) are given correctly by
((y2 + z2)ωx - xyωy - xzωz, -yxωx + (z2 + x2)ωy - yzωz, -zxωx - zyωy + (x2 + y2)ωz).
Do this both by working with components and by using the so-called BAC — CAB rule, that is A × (B × C) = B(A • C) — C(A • B).
(Taylor 10.19)
[do the thing]
Consider the bead threaded on a circular hoop rotating about its vertical diameter, working in a frame that rotates with the hoop. Find the equation of motion of the bead, and check that your result agrees with
θ" = (ω2cosθ - g/R)sinθ.
(Taylor 9.17)
As seen in a frame rotating with the hoop, there are five forces on the bead. The first three, all of which act in the plane of the hoop, are the bead’s weight mg, the centrifugal force Fcf = mω2Rsinθ ρ-hat, and the normal force N (actually the component of the normal force in the plane of the hoop). The other two are the Coriolis force Fcor, and the component of the normal force normal to the hoop. Since these last two both act normal to the hoop, they cancel one another and need not concern us further. The bead can move only in the tangential direction, and its equation of motion is matang = Ftang, or mRθ" = Fcfcosθ - mgsinθ = (mω2Rsinθ)cosθ - mgsinθ, whence θ" = (ω2cosθ - g/R)sinθ
A massless spring (force constant k1) is suspended from the ceiling, with a mass m1 hanging from its lower end. A second spring (force constant k2) is suspended from m1, and a second mass m2 is suspended from the second spring's lower end. Assuming that the masses move only in a vertical direction and using coordinates y1 and y2 measured from the masses' equilibrium positions, show that the equations of motion can be written in the matrix form My" = —Ky, where y is the 2 x 1 column made up of y1 and y2. Find the 2 x 2 matrices M and K.
(Taylor 11.2)
M = [m1 0]
[0 m2]
K = [k1+k2 -k2]
[ -k2 k2 ]
A rigid body consists of three masses fastened as follows: m at (a, 0, 0), 2m at (0, a, a), and 3m at (0, a, -a). (a) Find the inertia tensor I. (b) Find the principal moments and a set of orthogonal principal axes.
(Taylor 10.35)
a) I = ma2 [10 0 0]
[ 0 6 1]
[ 0 1 6]
b) λ1 = 10ma2, λ2 = 7ma2, λ3 = 5ma2
e1 = (1, 0, 0), e2 = (1/√2)(0, 1, 1), e3 = (1/√2)(0, 1, -1)
Show that the inertia tensor is additive, in this sense: Suppose a body A is made up of two parts B and C. (For instance, a hammer is made up of a wooden handle wedged into a metal head.) Then IA = IB + IC. Similarly, if A can be thought of as the result of removing C from B (as a hollow spherical shell is the result of removing a small sphere from inside a larger sphere), then IA = IB - IC.
(Taylor 10.20)
[prove it for diagonal and off-diagonal elements]