I need some help on a homework question. A U-shaped tube is partially filled wit

Question

I need some help on a homework question.

A U-shaped tube is partially filled with water, as shown.

A. Show that if the water is displaced a small distance x from equilibrium the water will execute simple harmonic motion. Hint what     time of force causes SHM? Find the force of the water in the tube.

B. Find an expression for the period of the oscillation. Assume that the tube diameter is d, and the total length of the water in the          tube is L.

I know that for anything to have SHM it needs to act like a restoring force like

F=-kx

However what im unsure of is, do I need to use F=?gh and F=PA along with F=ma  ?

I also know that eventually I should have a differential equation where x=cos(?t) is the solution to the equation. Im just not sure where Im going wrong at.

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 time of force causes SHM? Find the force of 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 the water in the tube is L. I know that for anything to have SHM it needs to act like a restoring force like F=-kx However what im unsure of is, do I need to use F=?gh and F=PA along with F=ma ? I also know that eventually I should have a differential equation where x=cos(?t) is the solution  to the equation. Im just not sure where Im going wrong at.

Explanation / Answer

P = p g x   where p = density, P = pressure, and x height of column

P2 - P1 = 2 p g x     difference in pressures

F = -2 p g A x    force on liquid of length L

(similar to F = - k x      for spring)

M = p L A    mass of liquid

F = M a = M d^2 x / dt^2

p L A d^2 x / dt^2 = -2 p g A x

d^2 x / dt^2 + 2 g x / L = 0

This is similar to d^2 x / dt^2 + w^2 x = 0    where w^2 = k / m for SHM

w = (2 g / L)^1/2

w = 2 pi f = 2 pi / T

T = 2 pi / w = 2 pi (L / (2 g))^1/2

I don't think you need d

Units for T = (cm / (cm / sec^2))^1/2 = sec

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