Classical mechanics exam
1-Note: The following problems are independent of one another! Parts a & b are about relativity. Part c is about rigid body rotation!
The 3 kilometer long Stanford Linear Accelerator accelerates electrons (rest energy mec2 = 0.5 MeV) to energy 50 GeV (50 109 eV). Use relativistic kinematics to calculate the speed of the electrons at the end.
A free neutron (rest energy 939.6 MeV) is unstable & decays into a proton (rest energy mpc2 = 938.3 MeV) & an electron (rest energy mec2 = 0.5 MeV). Use relativistic kinematics to calculate the energy released in this process.
See Figure. A pendulum consists of a uniform rod of length & mass m. It is suspended in a vertical plane by one end. At the other end, a uniform disk of mass M is attached so that it can rotate freely in its own plane, which is the vertical plane. The rod & the disk each have negligible thickness. Set up the equations of motion for this system in the Lagrangian formalism.
2-Parts a-c concern a particle of mass m moving in one dimension under the influence of a time & position dependent force given by F(x,t) = (kx-2)e-t/τ, where k & τ are positive constants.
Set up the Lagrangian & derive Lagrange’s Equation of motion.
Set up the Hamiltonian & derive Hamilton’s Equations of motion. Show that these are equivalent to the equation of motion obtained in part a.
Compare the Hamiltonian & the total mechanical energy. Are these the same? Is energy conserved for this system?
See Figure. Parts d-f concern a plane pendulum of mass m attached to a massless, rigid rod. After the pendulum is set into motion, the length is shortened at a constant rate α. That is, (d/dt) = - α. At time t = 0, the rod length is 0. The suspension point remains fixed. (Hint: The condition on (d/dt) is a time-dependent constraint which fixes the time dependence of ).
How many degrees of freedom are there for this system? Set up the Lagrangian & derive Lagrange’s Equation of motion.
Set up the Hamiltonian & derive Hamilton’s Equations of motion. Show that these are equivalent to the equation of motion obtained in part d.
Compare the Hamiltonian & the total mechanical energy. Are these the same? Is energy conserved for this system?
3-See Figure. A massless ideal spring of relaxed length b & spring constant κ connects two particles of masses m1 & m2. The system rests on a frictionless table & may oscillate ( is it’s time dependent length while it is oscillating) AND rotate in the plane of the table (θ is the angle of rotation in the plane).The figure looks down on the plane of the table.
How many degrees of freedom are there for this system? Set up the Lagrangian for this system & derive the Lagrange Equations of motion.
Set up the Hamiltonian for this system & derive Hamilton’s Equations of motion.
Compare the Hamiltonian & the total mechanical energy. Are these the same?
Is energy conserved for this system? Are there any other constants of the motion (physical quantities which are conserved)?
4-A plane triatomic molecule consists of equal masses m at the vertices of a triangle. If the masses were at rest, the masses would be on an equilateral triangle of side b. The molecule is held together by forces that are harmonic for small oscillations & the force constants are each equal to k. Consider motion in the plane of the molecule only.
Write expressions for the kinetic energy, the potential energy, & the Lagrangian of the system.
Derive the equations of motion using Lagrange’s equations.
Assume small oscillations. Set up & solve the solve the secular equation for the normal mode eigenfrequencies & eigenvectors. How many normal modes are there? Do any of them correspond to ω = 0?
Answer the following question using mostly WORDS, in complete, grammatically correct
English sentences! You may also make a sketch if that helps your discussion.
Qualitatively discuss the motion of the 3 masses in each of the normal modes.
