Question

Oscillations Problem

Answer 1

A body of uniform cross - sectional area \(A = 1~cm^2\) and of mass density \(\rho = 0.8 \frac{ g }{ cm^3 }\) floats in a liquid of density \(\rho _0 = 1 \frac{ g }{ cm^3 }\) and equilibrium displaces a volume \(V = 0.8 cm^3\). Show that the period of small oscillations about the equilbrium postion is given by \[\tau = 2 \pi \sqrt{\frac{ V }{ gA }}\] where g is the gravitational field strength. Also determine the value of \(\tau \).

Answer 2

Have fun :-)

Answer 3

Do you need help, or do you know how to solve this problem?

Answer 4

Hey Vincent :), I have already solved it, I sometimes like to put questions up so others can give it a go, if no one answers it or wants to take a shot at it I will post my solution, thanks for taking it as a concern!

Answer 5

Ok, that's what I was thinking, so I will not write down the solution straight away ;-)

Answer 6

\[\sum F = ma \\~\\\implies xA\rho_0g = -V\rho_0\ddot{x}\\~\\\implies \ddot{x} +\dfrac{Ag}{V}x = 0\]

Answer 7

here is my reasoning:
at Equilibrium, we have this situation:

we can write:
\[\Large g{\rho _o}{V_D} = g{\rho _B}{V_B} \Rightarrow \frac{{{\rho _o}}}{{{\rho _B}}} = \frac{{{V_B}}}{{{V_D}}}\]
where the subscripts \(B,\;D\) stands for body, and displaced
now, let's consider a little displacement \(z\) from the position of Equilibrium, due to the little oscillations