A U-tube open at both ends to atmospheric pressure P0 is filled with an incompressible fluid of density ρ . The cross-sectional area A of the tube is uniform and the total length of the column of fluid is L . A piston is used to depress the height of the liquid column on one side by a distance x0 , and then is quickly removed. What is the frequency of the ensuing simple harmonic motion? Assume streamline flow and no drag at the walls of the U-tube. (Hint: use conservation of energy). Express your answer in terms of L and acceleration due to gravity g (enter L for L and g for g).

f=

f = 1/(2*pi)*sqrt((2*g)/L)

To find the frequency of the ensuing simple harmonic motion, we can use the principle of conservation of energy.

Initially, when the piston depresses the height of the liquid column, it gains potential energy due to its displacement. This potential energy is converted into kinetic energy as the piston is removed, causing the liquid column to oscillate up and down.

The potential energy gained by depressing the liquid column is given by the formula:

PE = m * g * h
where m is the mass of the liquid column and h is the initial displacement.

The mass of the liquid column can be calculated using its volume and density:

m = ρ * A * L
where A is the cross-sectional area and L is the total length of the column.

Substituting this into the potential energy equation:

PE = ρ * A * L * g * h

At the highest point of the oscillation, all the potential energy is converted into kinetic energy. The kinetic energy at this point is given by:

KE = (1/2) * m * v^2
where v is the velocity.

Using the principle of conservation of energy, we can equate the potential energy and kinetic energy:

PE = KE
ρ * A * L * g * h = (1/2) * ρ * A * L * v^2

Simplifying the equation:

g * h = (1/2) * v^2

Since the motion is simple harmonic, the displacement can be related to the velocity using the equation:

v = ω * x
where ω is the angular frequency and x is the displacement.

Differentiating this equation with respect to time:

a = ω^2 * x
where a is the acceleration.

Acceleration can also be expressed as the second derivative of displacement with respect to time:

a = -(d^2x / dt^2)

Combining the two equations:

-(d^2x / dt^2) = ω^2 * x

This is the equation of simple harmonic motion, where ω is the angular frequency. Comparing the equations, we can see that:

ω^2 = g / h

The frequency of simple harmonic motion is given by:

f = ω / (2π)

Substituting the value of ω:

f = sqrt(g / h) / (2π)

Finally, substituting for h in terms of the given displacement x0:

f = sqrt(g / x0) / (2π)

Therefore, the frequency of the ensuing simple harmonic motion is sqrt(g / x0) / (2π).