at is difficult to model with N- of motion for the system from
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Please answer up to 3 subparts but if genrous enough what can answer :)
Transcribed Image Text:A large sphere containing a smaller sphere coupled to springs as shown in the diagram was used to
demonstrate complicated periodic motion that is difficult to model with Newton's laws directly. The
aim of this question is to find the equations of motion for the system from Lagrange's equations.
The physical parameters of the system are as follows: the two springs have a negligible mass, both
springs have the same spring constant k, both the large and small spheres have a mass m, the
gravitational force per kg is g. Let the centre of the large sphere and the centre of the small sphere
coincide when the system is at rest and in the equilibrium position. Define this to be x, = x2 = 0
where x, (t) and x2(t) are the positions of the centres of the large and small spheres respectively as
a function of time. The system is set in motion from some initial position x, (0) = a and x2 (0) = b.
(As a matter of interest the real lecture demonstration had k = 82 N/m, g = 9.8 N/kg and m =
3.5 kg with -a = b = 0.1 m)
(a) Find an expression for the total kinetic energy T of the system.
(b) Find an expression for the total potential energy V of the system.
(c) Hence determine the Lagrangian L of the system.
(d) By means of Lagrange's equations, find the differential equations that give the equations of
motion for the positions of the two spheres.
(e) By seeking trial solutions of the appropriate form, describe how the full equations of motion can
be found. Be sure to define any new constants you introduce. It is not essential to find the values
of these new constants in terms of the known constants of the problem.
(f) Note any peculiarities of your solutions especially where some combinations of k and m may not
have a solution that can be found by this method. Why can we not find solutions for these
combinations?
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