3 Home exercise sheet 3.1 Conservation Laws and Neother’s theorem

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3.1
Home exercise sheet
Conservation Laws and Neother’s theorem
Exercise 3.1: Disk rolling down an inclined plane
A hoop of mass m and radius R rollsl without slipping down an inclined plane. The
inclined plane has opening angle α and mass M , and itself slides frictionlessly along a horizontal surface.
Find the EOM of the system and solve them.
Exercise 3.2: Bead on a stick
A stick is pivoted at the origin and is arranged to swing around in a horizontal plane at
constant angular speed ω. A bead of mass m slides frictionlessly along the stick. Let r be
the radial position of the bead. Find the conserved quantity E given by
E=
X ∂L
i
∂ q̇i
q̇i − L
Explain why this quantity is not the energy of the bead.
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Exercise 3.3: Spherical Pendulum
A bead is free to slind along a frictionless hoop of radius R. The hoop rotates with
constant angular velocity ω around a vertical diameter.
1. Find the EOM for the angle θ shown.
2. What are the equilibrium positions?
3. What is the frequency of small oscillations about the stable equilibrium?
4. There is one value of ω that is rather special; what is it, and why is it special?
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