Problem:
In a dynamical system with two degrees of freedom, the kinetic energy
is 
,
and the potential energy is V = c + dq2, where a, b, c, and d
are constants. Show that the value of q2 as a function of time is
given by an equation of the form (q2-k)(q2+2k)2 = h(t-t0)2
where h, k, and t0 are constants.
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Problem:
A uniform hoop of mass m and radius r rolls without slipping on a fixed cylinder of radius R as shown in the figure. The only external force is that of gravity. If the hoop starts rolling from rest on top of the big cylinder, find by the method of Lagrange multipliers the point at which the hoop falls off the cylinder.
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Problem:
A particle is constrained to move on the surface of a sphere of radius R0 centered about the origin (0,0,0) in the usual Cartesian coordinates x,y,z. A two-dimensional harmonic oscillator potential is applied in the y-z plane: V(x,y,z)=(1/2)k(y2+z2). The Lagrangian may be written as
L=(1/2)mv2-(1/2)k(y2+z2)-l(r-R0),
where v=(dx/dt,dy/dt,dz/dt), r = (x2+y2+z2)1/2,
m is the mass of the particle, k is the spring constant and l
is the respective Lagrange multiplier for the constraint that r=R0.
r=(x,y,z)
is the particle’s position vector.
(a) Find the equations of motion for the system.
(b) For the initial conditions r(0)=(0,0,R0), v(0)=(0,wR0,0), find the complete solution for r(t) and
l, for all t.
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Problem:
A square consists of four equal length (2a) rods joined together by frictionless pivots. Rods AB and BC are uniform and of mass m, rod CD is massless while rod AD is massless except for a mass 2m concentrated at its midpoint. An impulse acts at point A directed along the line defined by AC. Find the ratio of the initial velocities of points C and A, i.e. (Vc/Va). (The moment of inertia of a uniform rod of mass m, length 2a, about a perpendicular axis through its mass center is (1/3)ma2.)
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Problem:
Six equal, uniform rods, fastened at their ends by frictionless pivots, form a regular
hexagon and lie on a frictionless surface. A blow is given at a right angle to the
midpoint of one of them, so that it begins to slide with velocity u.
Show that
the opposite rod begins to move with velocity v=u/10.
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Problem:
A bead slides without friction on a wire in the shape of a
cycloid with equations
x=a(q-sinq),
y=a(1+cosq), 0<q<2p.
Use the Lagrangian method to find the equation of motion and show that the equation is of
the form
(d2u/dt2)+(g/4a)u=0, where
u=cos(q/2).
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Problem:
A bead of unit mass m=1 slides from rest at a point O(x=0, y=0)
without friction on a wire in a vertical plane (x,y) to a point A(x=0,
y=2pb) under the influence of gravity along
the positive x direction. If the total time taken is a minimum,
(a) prove that the curve is of the form
x=b(1-cosq),
y=b(q-sinq).
(b) Find the reaction force by the curve on the bead as a function of
q.
Solution:
