More Problems

Problem 1:

Two-dimension polar coordinates have basis vector er and eφ.
(a)  Draw a figure that shows the Cartesian basis vectors i and j, polar basis vectors er and eφ for an arbitrary position vector in the first quadrant.
(b)  Recall that the Cartesian infinitesimal line element is dr = dx i + dy j.
Express the infinitesimal line element in terms er and eφ.
(c)  Derive an expression for the velocity v in terms of polar basis vectors er and eφ.

Solution:

Problem 2:

Two masses, m1 and m2, lie on a frictionless, horizontal table.  They are connected by a spring of spring constant k.
(a)  Write down the Lagrangian for this system in the form

image

where xi denotes the displacement of mass mi from its equilibrium position.
(b)  Find the equations of motion.
(c)  Find the normal mode frequencies.

Solution:

Problem 3:

A particle of mass m moves in one dimension under the influence of a force F(x,t) = (k/x2)exp(-t/τ), where k and τ are positive constants.  Compute the Lagrangian and the Hamiltonian, and discuss whether energy is conserved in this system.

Solution:

Problem 4:

A mass point glides without friction on a cycloid defined by x = a(θ - sinθ), y = a(1 + cosθ),
with 0 ≤ θ ≤ 2π.
A uniform gravitational field g points in the negative y-direction.
The potential energy of the particle constrained to move on the given cycloid is symmetric about
θ - π.  Choose new coordinate that better reveal that symmetry.
Let 2φ = θ - π,  -π/2 ≤ φ ≤ π/2.  Then x = a(2φ  + π + sin2φ), y = a(1 - cos2φ).
The bead moves on a trajectory s with elements of arc length ds.
ds = (dx2 + dy2)½ = ((∂x/∂φ)2 + (∂y/∂φ)2)½dφ.
Write down the Lagrangian for this system in terms of the coordinate s and the velocity ds/dt with the condition s = 0 at φ = 0.

Solution:

Problem 5:

Consider the systems below shown in its equilibrium position below.

image
The springs hasspring constant k.
Consider only motion in the plane of the figure.
For small displacements of the masses from their equilibrium positions, the Lagrangian can be written as
L = ½∑ij[Tij(dqi/dt)(dqj/dt) - kijqiqj
with Tij = Tji,  kij = kji, where the qi are generalized coordinates specifying a displacement from equilibrium.
Choose appropriate generalized coordinates qi and determine all coefficients Tij and  kij in the Lagrangian.

Solution: