Assignment VII, PHYS 301 (Classical Mechanics)Spring 2013

Due 3/1/13 at start of class

1. Recall the following setup from your first exam shown below. As before, each springhas equilibrium length x◦ = D/2; the upper spring has spring constant k1 and thelower spring has spring constant k2. Once again, this wedge is on Earth. Before, wewere just looking for a stability condition. Now, we want an equation of motion.

a) Again using ` as your single coordinate marking distance from the lower cornerof the wedge, write down the Lagrangian for this system.

b) Use the Euler-Lagrange equation to find a differential equation for `. Write thisequation with all terms involving ` on the left hand side, and all constant termson the right hand side.

c) You should notice that the left hand side looks like the equation for a simpleharmonic oscillator, but the right hand side is not zero like we have for our usualsimple harmonic oscillators. This suggests that the oscillation does not centeraround ` = 0. Try applying a change of variables s = ` − xe where xe is theequilibrium position of the oscillator. After applying this change of variables,rewrite the differential equation. Then, find s(t) (you will have two unspecifiedconstants; just leave those as unspecified constants for now). Use this expressionto find `(t).

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2. The hot-air balloon pendulum! A hot air balloon accelerates vertically with constantacceleration a near the surface of the earth. On the bottom of the balloon’s passengerbasket, someone cleverly mounted a simple pendulum with arm-length ` and bob-massm. You may assume that the arm is massless.

a) Form the Lagrangian for this system (assuming you stay close to the surface ofthe Earth). (The Lagrangian will depend on t and the angle φ with respect tothe vertical, as well as constants).

b) Find φ(t) for small oscillations. (Hint for those of you in GR: equivalence princi-ple!)

c) Rewrite the Lagrangian for this system without the assumption that you stayclose to the Earth. (i.e. g is no longer a constant!) (Don’t worry; I won’t makeyou solve it).

3. The two springs shown below are constrained to move along a single horizontal direc-tion. Choose as generalized coordinates the displacements of the springs from theirequilibrium lengths.

a) Write the Lagrangian for the system.

b) Use the Euler-Lagrange equations to find two differential equations that describethe motion. (They will be coupled second order differential equations. No needto solve them at this point, but if you want to show off, go ahead and do it. We’lltalk about systems like this in a couple more chapters).

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4. We did problem 4.36 earlier this semester in class. Go back to that problem andthe associated figure (figure 4.27). We mentioned that this system only has 1 de-gree of freedom. Find the Lagrangian for this system, using θ as the one coordinate.(Trigonometry will be of some help here).

5. A sphere of radius a and mass M is constrained to roll without slipping (don’t forgetthe rotational energy!) on the lower half of the inner surface of a hollow cylinderof inside radius R. You may assume the cylinder remains stationary with respect toEarth.

a) Determine the Lagrangian function.

b) Write down the equation of motion. (You do not need to solve the differentialequation; just get the differential equation).

c) What is the frequency (NOT angular frequency) of small oscillations?

6. Figure 7.14 in your text (pg. 286) shows a simple pendulum of mass m and length` whose point of support is attached to the edge of a wheel of radius R rotatingat a fixed angular velocity ω. At t = 0, the point P is level with O on the right.Write down the Lagrangian and show the equation of motion takes the form `φ̈ =−g sinφ+ ω2R cos(φ− ωt). Confirm this makes sense for ω = 0.

These last two problems are actually part of homework 8. However, since they are com-putational in nature, they may take some of you longer than your average homeworkproblem and I wanted to get them into your hands earlier so you can get a head-starton them. They will be due with the rest of HW 8 the Friday after Spring Break (alongwith the rest of hw8 that will be assigned next week). This also lets you get a head-start on the next homework assignment so you can, in principle, have more of yourSpring Break free for other activities.

7. Do problem 7.44 in your text. (The previous problem is essentially problem 7.29).

8. Similar to the above, replot φ(t) with the new conditions with g = ` = 1 but ω = 3 andR = 2. Let initial conditions be φ = 2 and φ̇ = 0. Plot your solution for 0 < t < 10.Comment on your plot.

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