Download 10 Class exercise sheet

10
Class exercise sheet
Exercise 10.1: Rotating tube of fluid
Consider a cylinder filled with a liquid, rotating with an angular frequency ω about it’s
symmetry axis ẑ.
We consider the steady state where the fluid is stationary in the rotating frame. The exercise
will be done for a fluid element making the calculations simpler
1. ~r is the position vector of a fluid element, write it as a function of ρ and z(ρ).
2. Write the EOM for a fluid element, labeling the force per unit mass on the fluid element
as f~.
3. Find the direction and size of f (i.e. n̂(ρ) and |f~|). Remember that the fluid is in the
steady state.
4. Find d~r (just the differential of ~r) and explain why n̂ · d~r = 0.
5. Using the last item find the shape of the surface (i.e z(ρ)).
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Exercise 10.2: Oscilator in a magnetic field (without the field...)
A mass is fastened to a frame by springs with elastic constants κ1 and κ2 . The frame
rotates in it’s own plane with an angular velocity Ω. The mass moves in the plane of the
frame. Show that this problem is equivalent to a 2D harmonic oscillator in a magnetic
potential.
Hint: Choose the magnetic potential as A = 12 B(−y, x, 0)
Exercise 10.3: Compton generator
The Compton generator is a simple device to demonstrate the Coriolis force due to the Earth
rotation. It consists of a narrow glass tube bend into a closed ring that is completely filled
with water and some small particles to observe the motion of the water in the tube. Initially
the ring is horizontal and the water is stationary. The ring is then quickly rotated by 180◦
around its East-West diameter and stopped.
1. Find the rotation of the earth ω
~ and position of a particle ~r in the following coordinate
system. where W − E is the rotation axis.
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2. Find the acceleration due to the Coriolis force. What is the tangential acceleration?
3. Average over the ring to find the mean acceleration. This is done because the fluid is
incompressible.
4. Finally find the velocity of the water in the tube after the ring is rotated. You can
assume that the ring was rotated at a constant angular velocity φ = 2π
t, where τ is
τ
the period of revolution.
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