# Computing differential changes

1. Dec 4, 2005

### capslock

A glass hydrometer of mass m floats in water of density p. The area of the neck of the hydrometer is A. The resonant frequancy for the vertical oscillations of the hydrometer is given by

f = (1/2\pi) (Apg/m)^(1/2)

Calculuate the fractional change in the resonant frequancy when the temprature changes from 20 to 30 degrees C.

My effort:

(2f\pi)^2 = Apg/m
2ln(2f\pi) = ln(Apg/m)
2ln(2) + 2ln(\pi) + 2ln(f) = ln(A) + ln(P) + ln(g) - ln(m)

I tried to then differentiate but I got in a total kefuffel.

In the question it gives the 'coefficient of linear expansion' for glass and the 'coefficient of volume expansion for water.

2. Dec 5, 2005

### mezarashi

First off, I'm not familiar with hydrometers, but as for rates of change, the question is asking you to find something of the form

$$\frac{df}{dT} = g(f, T)$$

and of course hope that the equation turns out to be separable and thus easily solvable by integration over the range you need (i.e. T=20, T=30).

The information given to you, change in volume over time and change in length over time are: dV/dT and dx/dT.

These two variables V, and x are not in the original equation. You need to successively find relationships, for example:

$$\rho = \frac{m}{V}$$
$$\frac{d\rho}{dT} = -\frac{m}{V^2}\frac{dV}{dT}$$

Once you have the right relationships you can plug in.