- #1
RockenNS42
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1. Homework Statement
A U-tube has vertical arms of radii 'r' and '2r', connected by a horizontal tube of length 'l' whose radius increases linearly from r to 2r. The U-tube contains liquid up to a height 'h' in each arm. The liquid is set oscillating, and at a given instant the liquid in the narrower arm is a distance 'y' above the equilibrium level.
Show that the kinetic energy of a small slice of liquide in the horzional arm is given by
dk= (1/2)ρv2 * (∏r2)dx/ (1+x/l)2 (1)
where v is the velocity
ρ is denisty
l is the lengh from the centre of one arm to the centre of the other
x is the distance the slice is from the middle if the smaller arm
2. Homework Equations
k = 1/2mv2=1/2ρAlv2 (2)
3. The Attempt at a Solution
So at first I seen the dK and dx and thought to intergrate, but I got a really messy problem so I figured that was it. SO i thought Id prove it bit by bit
i got that the l in (2) is the 1/(1+x/l)2 from (1) but I am not sure how to go about proving this is the case...
A U-tube has vertical arms of radii 'r' and '2r', connected by a horizontal tube of length 'l' whose radius increases linearly from r to 2r. The U-tube contains liquid up to a height 'h' in each arm. The liquid is set oscillating, and at a given instant the liquid in the narrower arm is a distance 'y' above the equilibrium level.
Show that the kinetic energy of a small slice of liquide in the horzional arm is given by
dk= (1/2)ρv2 * (∏r2)dx/ (1+x/l)2 (1)
where v is the velocity
ρ is denisty
l is the lengh from the centre of one arm to the centre of the other
x is the distance the slice is from the middle if the smaller arm
2. Homework Equations
k = 1/2mv2=1/2ρAlv2 (2)
3. The Attempt at a Solution
So at first I seen the dK and dx and thought to intergrate, but I got a really messy problem so I figured that was it. SO i thought Id prove it bit by bit
i got that the l in (2) is the 1/(1+x/l)2 from (1) but I am not sure how to go about proving this is the case...