Oscillation of a Mass in Liquid (Densities and Cross-Sectional Area Known)

[SerenityBear]

Homework Statement

A body of uniform cross-sectional area A = 1 cm^2 and of mass density p = 0.8 g/cm^3 floats in a liquid of density p0 = 1 g/cm^3 and at equilibrium displaces a volume V = 0.8 cm^3. Show that the period of small oscillations about the equilibrium position is given by:

T = 2pi(V/gA)^(1/2)

Where g is the gravitational field strength.

Homework Equations

T = 2pi(m/k)^(1/2)
m = pV

The Attempt at a Solution

I think that, since the general solution for the period of an oscillation (in the relevant equations section) is very similar to the solution I'm looking for, that I'm probably trying to prove that what is under the square roots is equal to each other, that is, m/k = V/gA.

The mass of an object is m = pV. (I also found an equation stating that at equilibrium, the mass of an object floating in a liquid is m = p0V (density of the liquid times the volume of liquid displaced). I don't know which of these I want to use, or even how it's possible for them to both be true.)

But if you use either of those equations (let's say m = pV because right now, that makes more sense to me , then you can plug that value for mass into the m/k = V/gA equation, and you end up with p = k/gA. So, I feel like if I can prove that, then I have my solution, but I don't know where to go from here.

Some questions: What is the difference between m = pV and m = p0V, and which do I want to use? What is k in this problem? How is density related to gravity and cross-sectional area (or is it)? And, of course, am I on the right track at all? Thanks in advance! :)

 

[Astronuc]

One's approach is correct. Think about the form of the equation for the oscillation of a mass m on a spring with constant, k. The restoring force is proportional to the displacement, x, from equilibrium. Similar, the bouyant force is related to the volume of the mass in the water, and there is some equilibrium depth. Push the mass down by some small displacement, and when the push is removed the bouyant force restores the mass to it's equilibrium position, or perhaps it bobs up and down if the push is removed instanteously.

Note that the problem mentions small displacement, which implies an approximation of the restoring force, similar to kx.

 

[baba89]

Dear student,

You are definitely on the right track. Let's start by addressing your first question about the difference between m = pV and m = p0V. The equation m = pV represents the mass of the object, where p is the density of the object and V is the volume of the object. On the other hand, the equation m = p0V represents the mass of the liquid displaced by the object, where p0 is the density of the liquid and V is the volume of liquid displaced. In this problem, we are interested in the mass of the object, so we will use m = pV.

Next, let's talk about the constant k in the equation T = 2pi(m/k)^(1/2). This constant represents the spring constant, which is a measure of the stiffness of the spring. In this problem, we are dealing with an object floating in a liquid, so there is no spring involved. Therefore, we can ignore the constant k and focus on the relationship between mass, gravity, and cross-sectional area.

Now, to prove that m/k = V/gA, we can start by substituting the equation m = pV into the equation T = 2pi(m/k)^(1/2). This gives us:

T = 2pi(pV/k)^(1/2)

Next, we can use the equation V = m/p to substitute for V, giving us:

T = 2pi(p(m/p)/k)^(1/2)

We can simplify this to:

T = 2pi(m/kp)^(1/2)

Finally, we can use the equation g = kp to substitute for kp, giving us:

T = 2pi(m/g)^(1/2)

And since m = pV, we can substitute for m, giving us the final equation:

T = 2pi(V/gA)^(1/2)

This shows that the period of small oscillations about the equilibrium position is indeed given by T = 2pi(V/gA)^(1/2). I hope this helps clarify the problem for you. Keep up the good work!