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Jack21222
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(apologies for the formatting, mixing LaTeX and regular text looks ugly, but I don't know how to do it otherwise)
This is problem 3-7 from Thornton and Marion's Classical Dynamics.
A body of uniform cross-sectional area A = 1 cm2 and of mass density \rho=0.8 g/cm3 floats in a liquid of density \rho0=1 g/cm3 and at equilibrium displaces a volume V = 0.8 cm3. Show that the peroid of small oscillations about the equilibrium position is given by:
\tau = 2\pi \sqrt{V/gA}
where g is the gravitational field strength. Determine the value of \tau
\tau = 2\pi\sqrt{k/m} (equation 3.13 in the book)
k\equiv -(dF/dx)
It appears to me that if I can show V/gA = m/k or k = mgA/V, then the equation in the problem would be equal to equation 3.13 quoted above. I tried coming up with an equation for the restoring force F in terms of mg A and V, and then take the x-derivative of k, but it doesn't equal mgA/V.
I'll be using p = rho from here, because I don't want to mix LaTeX and regular text, and p has no other meaning.
The restoring force I came up with was F = Vp0g - mg where m is the mass of the block.
In terms of x, I get F = Axpg - mg where x is the depth of the submerged block.
Taking the x-derivative of F, from the definition of k, I get Apg, which isn't equal to mgA/V
Any suggestions?
Homework Statement
This is problem 3-7 from Thornton and Marion's Classical Dynamics.
A body of uniform cross-sectional area A = 1 cm2 and of mass density \rho=0.8 g/cm3 floats in a liquid of density \rho0=1 g/cm3 and at equilibrium displaces a volume V = 0.8 cm3. Show that the peroid of small oscillations about the equilibrium position is given by:
\tau = 2\pi \sqrt{V/gA}
where g is the gravitational field strength. Determine the value of \tau
Homework Equations
\tau = 2\pi\sqrt{k/m} (equation 3.13 in the book)
k\equiv -(dF/dx)
The Attempt at a Solution
It appears to me that if I can show V/gA = m/k or k = mgA/V, then the equation in the problem would be equal to equation 3.13 quoted above. I tried coming up with an equation for the restoring force F in terms of mg A and V, and then take the x-derivative of k, but it doesn't equal mgA/V.
I'll be using p = rho from here, because I don't want to mix LaTeX and regular text, and p has no other meaning.
The restoring force I came up with was F = Vp0g - mg where m is the mass of the block.
In terms of x, I get F = Axpg - mg where x is the depth of the submerged block.
Taking the x-derivative of F, from the definition of k, I get Apg, which isn't equal to mgA/V
Any suggestions?