Frequency of girl bobbing in swimming pool

[kevlar94]

Homework Statement

A girl with mass m kg steps into her inflatable ring with horizontal cross sectional area Am^2 and jumps into the pool. After the first splash, what is the frequency of the girl bobbing up and down?

Homework Equations

I assume that we need the extra force,F_e, after the buoyant force and the weight cancel. Archimedes F_b = mg

We can then use Newton 2, F=ma, where a=x".
x" = sqrt(F_e/m)

f=1/[2π√(F_e/m)]

The Attempt at a Solution

The above ω will be for the frequency.

I am not sure if the above is right and I do not know how to solve for the F_e

Thanks!

 

[collinsmark]

Hello kevlar94,

Welcome to Physics Forums!

Homework Statement

A girl with mass m kg steps into her inflatable ring with horizontal cross sectional area Am^2

Just so we are both on the same page, do you really mean that the cross sectional area is a function of the mass squared? Maybe the problem statement is written that way, but I just want to be sure.

In other words, do you really mean that

$$\mathrm{Area} = Am^2$$

where $m$ is the mass and $A$ is an some constant of proportionality?

and jumps into the pool. After the first splash, what is the frequency of the girl bobbing up and down?

Homework Equations

I assume that we need the extra force,F_e, after the buoyant force and the weight cancel. Archimedes F_b = mg

Well, you can say F_b = mg if nothing is accelerating, and is in static equilibrium. But that's not the case when the girl is bobbing up and down.

Perhaps you mean to say that F_e = F_b - mg?

We can then use Newton 2, F=ma, where a=x".
x" = sqrt(F_e/m)

Where did the square root come from? Newton's second law doesn't contain a square root.

$$\sum_i \vec F_i = m \ddot{z}$$

(since the motion in this problem is always in the up/down direction, I chose to use the variable $z$ to represent the position. You could just as easily use the variable $x$ to represent position if you want to though.)

f=1/[2π√(F_e/m)]

You'll have to show me where that came from.

The Attempt at a Solution

The above ω will be for the frequency.

I am not sure if the above is right and I do not know how to solve for the F_e

There are two forces involved. The weight of the girl and the buoyant force. You've already figured out the weight is mg.

The buoyant force is equal to the weight of the water that is displaced. The weight of the water is proportional to g and the density of water ρ. It's also proportional to the volume of the water that is displaced.

The cross sectional area is already given in the problem statement. You just need to throw in the vertical displacement to find the volume of displaced water.

Plug those back into Newton's second law and you'll get an equation containing both $z$ and $\ddot{z}$: an ordinary, second order differential equation that you can solve.

 

[kevlar94]

Thanks for the help.

Just so we are both on the same page, do you really mean that the cross sectional area is a function of the mass squared? Maybe the problem statement is written that way, but I just want to be sure.

In other words, do you really mean that

$$\mathrm{Area} = Am^2$$

where $m$ is the mass and $A$ is an some constant of proportionality?

Sorry, the prompt says the girl is 25kg and the inflatable ring has a horizontal cross-sectional area of 0.7m^2.

Well, you can say F_b = mg if nothing is accelerating, and is in static equilibrium. But that's not the case when the girl is bobbing up and down.

Perhaps you mean to say that F_e = F_b - mg?

Where did the square root come from? Newton's second law doesn't contain a square root.

Yes, that is what I meant. I skipped a step I should have mentioned.

The square root is from the formula for frequency using k(from hookes law) from my book.

So F_r = F_b - mg = ρ(displaced water)(V)g

I start with the assumption that the ring goes dx distance into the water. Which results in dV=Adx

dF_r=ρ*g*A*dx

Since I am using hookes law for a linear oscillator, F=kx or dF=kdx

so dF_r = kdx= ρ*g*A*dx which gives a k value of ρ*g*A -- when the dx cancels

Using k I can solve for ω using √(k/m) and then solve for frequency using ω=2π*f

Using the above values gives ω= √(9800*.7)/25 = 16.56

f=16.56/(2π) = 2.637Hz

Does that look correct?

 

[collinsmark]

Sorry, the prompt says the girl is 25kg and the inflatable ring has a horizontal cross-sectional area of 0.7m^2.

Oh, 'm' is meters (not mass). Okay, I understand now.

Yes, that is what I meant. I skipped a step I should have mentioned.

The square root is from the formula for frequency using k(from hookes law) from my book.

Ah, Hooke's law. I never thought of that. The buoyant force is proportional to the displacement. So yes, Hooke's law and associated equations will work just fine. That should save you from having to solve the differential equation -- essentially from re-deriving the ω = √(k/m) formula.

So F_r = F_b - mg = ρ(displaced water)(V)g

I start with the assumption that the ring goes dx distance into the water. Which results in dV=Adx

dF_r=ρ*g*A*dx

Since I am using hookes law for a linear oscillator, F=kx or dF=kdx

so dF_r = kdx= ρ*g*A*dx which gives a k value of ρ*g*A -- when the dx cancels

Using k I can solve for ω using √(k/m) and then solve for frequency using ω=2π*f

Using the above values gives ω= √(9800*.7)/25 = 16.56

f=16.56/(2π) = 2.637Hz

Does that look correct?

That's what I got (out to the first three significant figures anyway). Good job.

 

[asteriatic]

I would approach this problem by first defining the variables and assumptions. The mass of the girl (m) and the cross-sectional area of her inflatable ring (A) are given. We can assume that the girl's motion in the water is simple harmonic motion, where the restoring force is the buoyant force (F_b) and the weight (mg) acting in opposite directions.

Using Newton's second law, we can set up the equation F_b - mg = ma, where a is the acceleration of the girl's motion in the water. Since we are assuming simple harmonic motion, we know that a = -ω^2x, where ω is the angular frequency and x is the displacement from equilibrium.

Next, we can use the equation for buoyant force, F_b = ρVg, where ρ is the density of the fluid (water), V is the volume of the displaced fluid, and g is the acceleration due to gravity. We can substitute this into our equation and solve for the angular frequency ω.

ω = √(g/ρV)

To find the frequency (f), we can use the relationship f = ω/2π.

f = 1/[2π√(g/ρV)]

Therefore, the frequency of the girl bobbing up and down in the swimming pool would depend on the density of the water, the volume of the displaced water, and the acceleration due to gravity. It would also be affected by the girl's mass and the cross-sectional area of her inflatable ring, as these factors determine the buoyant force.

Further research and experimentation may be needed to accurately determine the frequency of the girl's motion, as there may be other factors at play such as drag and surface tension. However, this approach provides a starting point for understanding the relationship between the girl's motion and the given variables.