Please explain throughly! Short answer. Show all work, starting with an original

Question

Please explain throughly!

Short answer. Show all work, starting with an original equation using variables, for maximum credit. A U-shaped tube is partially filled with water, as shown. Show that if the water is displaced a small distance x from equilibrium, the water will execute simple harmonic motion. Hint: what type of force causes SHM? Find the force on the water in the tube. Find an expression for the period of the oscillation. Assume that the tube diameter is d, and the total length of water in the tube is L.

Explanation / Answer

a U-shaped tube with water is shaken so that water rises and falls in each of its branches. The higher water rises in one branch, the lower it falls in the other branch, and the more pressure difference is generated to push it back. This pressure difference is again proportional to the height of rising/falling.This will led to SHM in U shaped tube.

Regarding the time period part

if the displacement is x then the level in one part is 2x more than than other part,however displacment from equilibrium position is x only as is evident in your figure.the excess pressure in that limb is

p=rho*g*x

F=p*A

F=rho*g*A*x

=>k=rho*g*A=rho*g*pi*r^2..

m=rho*pi*r^2*L

k/m=g/L.

T=2*pi*sqrt(L/g).

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