Find the normal, longitudinal modes of vibration for three
masses connected by identical springs of spring constant k. The masses are
collinear. The end masses have mass m, while the inner mass has mass 2m.
(a) Calculate the normal modes of the system.
(b) Describe the relative motion of the particles for each normal mode.
We can find the ω2 from det(kij-ω 2Tij) = 0. For a system with n degrees of freedom, n characteristic frequencies ωa can be found. Our system has 3 degrees of freedom.
2(k-ω2m)3 - 2k2(k-ω2m) = 0.
Solution 1: k-ω2m = 0, ω = (k/m)½.
Solution 2: k-ω2m ≠ 0, then (k-ω2m)2 = k2, (k-ω2m) = ±k.
For (k-ω2m) = +k, ω = 0.
Solution 3: (k-ω2m) = -k, ω = (2k/m)½.
(b) The displacements for each mode are determined from the equations of motion.
Solution 1: ω = (k/m)½.
Equation 1 yields kA2 = 0, equation 2 then yields kA1 + kA3 = 0.
A1 = - A3, A2 = 0, the central mass is stationary, m1 and m3 move in opposite directions with equal amplitudes.
Solution 2: ω = 0.
A1 = A2 = A3, translation of the CM, no relative motion.
Solution 3: ω = (2k/m)½.
Equation 1 yields -kA1 - kA2 = 0, A1 = -A2.
Equation 3 yields -kA2 - kA3 = 0, A3 = -A2.
A1 = A3 = -A2, the central mass move in a direction opposite to the direction of the outer masses. All masses oscillate with equal amplitudes.
A triple pendulum consists of masses αm, m and m attached to a single light
string at distances a, 2a, and 3a respectively from its points of suspension.
Consider only motion in a plane.
(a) Determine the value of α such that one of the normal frequencies of the system will equal the frequency of a simple pendulum of length a/2 and mass m. You may assume the displacements of the masses from equilibrium are small.
(b) Find the mode corresponding to this frequency and sketch it.
(b) Let λ = ω2a/g. Then
|(α + 2)(1 - λ)||-2λ||-λ|
|-2λ||2(1 - λ)||-λ|
|-λ||-λ||(1 - λ)|
|α + 2||4||2|
= 0 --> α = 2.
(b) The mode corresponding to this frequency is found from
|α + 2||4||2|
A3 = -2A1, A2 = 0.
Consider an ideal gas of N particles in a cylinder with a piston so that the
volume and pressure may change. Suppose that each particle has f = 5 quadratic degrees
of freedom in its energy. Consider the three paths in a PV diagram shown. Suppose ways 1
and 3 are straight lines on the PV diagram and way 2 is an adiabatic process.
(a) How much work W is done on the gas for ways 1, 2, and 3?
(b) How much heat Q is transferred to the gas for ways 1, 2, and 3?
(c) What is the change in internal energy ΔU for ways 1, 2, 3?
(d) What is the change in entropy ΔS for way 3?
Consider a ring of radius, r, with 4 identical point particles of mass m
interconnected by identical springs with spring constant k to their nearest
neighbors. The particles move without friction. The interconnected springs can
be viewed as causing harmonic oscillations.
(a) Determine the number of normal modes of oscillations, and establish the Lagrangian.
(b) Find the frequencies for small oscillations and describe the corresponding eigen-vibrations.
L = ½∑ij[Tij(dqi/dt)(dqj/dt) - kijqiqj]
with Tij = Tji, kij = kji.
Here Tii = m, Tij(i ≠ j) = 0, i, j = 1, 2, 3, 4, (cyclic).
kii = 2k, kij = -k if j = i ± 1, kij = 0 otherwise.
Equations of motion: d/dt(∂L/∂(dqi/dt)) - ∂L/∂qi = 0.
Solutions of the form qj = Re(Ajeiω t) can be found. For a system with n degrees of freedom, n characteristic frequencies ωα can be found. Some frequencies may be degenerate.
For a particular frequency ωα we solve ∑j[kij - ωα2Tij]Ajα
= 0 to find the Ajα.
Since it is difficult to evaluate the determinant of a 4 x 4 matrix, we find the solutions of this system of coupled equations using physical insight.
(b) Equation of motion for mass i:
(-kqi-1 + 2kqi - kqi+1) + md2qi/dt2 = 0, i,j = 1, 2, 3, 4, (cyclic).
1.) ω1 = 0, A1 = A2 = A3 = A4,
∑jkij = 2k - 2*k = 0.
All other modes can have no net angular momentum.
2.) Assume A1 and A3 are fixed. A1 = A3 = 0, A2 = -A4.
The equation of motion for i = 2 is k21A1 + k22A2 + k23A3 - ω22T22 A2 = 0,
2k - ω22m = 0, ω22 = 2k/m.
3.) Assume A2 and A4 are fixed. A1 = -A3 = 0.
For this mode ω32 = 2k/m. Mode 2 and 3 are degenerate. Any linear combination of these modes is also a normal mode. For example, the mode with A2 = A3 = 0, A1 = A4, , A1 = - A2 is a linear combination of modes 2 and 3 and is not linearly independent.
4.) Assume A1 = A3, A2 = A4, A1 = - A2.
The equation of motion for i = 2 is k + 2k + k - ω22m = 0, ω42 = 4k/m.
For the eigen-vibrations we have:
1.) qi = vt. (uniform rotation)
2.) q1 = q3 = 0, q2 = C2cos(ω2t + φ2), q4 = -C2cos(ω2t + φ2).
3.) q2 = q4 = 0, q3 = C3cos(ω3t + φ3), q1 = -C3cos(ω3t + φ3).
4.) q1 = q3 = C4cos(ω4t + φ4), q2 = q = -C4cos(ω4t + φ4).
The Ci and φi depend on the initial conditions. The mode with the highest frequency is mode 4 (non-degenerate) and the mode with the lowest frequency is mode 1 (non-degenerate).
A metal block of mass m and specific heat c with temperature Tb is placed into the ocean which has temperature T. Assume Tb > T. What is the total change in the entropy, ΔStotal, for the system? Express your answer in terms of m, c, Tb, and T.