Deriving a Differential Equation

[JNBirDy]

Homework Statement

Assuming that the rate of the mass (m) of an object is proportional to its total surface area, derive a D.E for the rate of change in the mass. Arbitrary constants may be included.

The objects in question are expanding peat discs. The discs absorb water through their surface which causes an increase in height (but the radius and density remain constant).

Homework Equations

None

The Attempt at a Solution

No idea...

Not even sure where to start? dm/dt = ...?

 

[cepheid]

Hello JNBirDy

The disk is essentially a very thin cylinder with a circular base having area A and with a height (or thickness) h. Only h is increasing with time. The area A remains constant. However, the volume of the disk depends on both A and h. As a result, the volume is increasing with time.

Since the density of the disk remains constant, the mass of the disk is just going to be proportional to the volume of the disk. So, the rate at which the mass is increasing with time just depends on the rate at which the volume is increasing with time. You can write down an expression for the latter.

Does that help?

 

[JNBirDy]

Hello JNBirDy

The disk is essentially a very thin cylinder with a circular base having area A and with a height (or thickness) h. Only h is increasing with time. The area A remains constant. However, the volume of the disk depends on both A and h. As a result, the volume is increasing with time.

Since the density of the disk remains constant, the mass of the disk is just going to be proportional to the volume of the disk. So, the rate at which the mass is increasing with time just depends on the rate at which the volume is increasing with time. You can write down an expression for the latter.

Does that help?

Hmm.. still not sure if I'm fully understanding it.

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The mass of the disk increases at a rate proportional to the volume as it increases in time. If I let r represent the growth rate, and dV/dt represent the change in volume w.r.t., then I get:

dm/dt = r(dV/dt)

This doesn't seem right to me?

 

[cepheid]

Hmm.. still not sure if I'm fully understanding it.

---

The mass of the disk increases at a rate proportional to the volume as it increases in time. If I let r represent the growth rate, and dV/dt represent the change in volume w.r.t., then I get:

dm/dt = r(dV/dt)

This doesn't seem right to me?

Step back for a second: let's say I gave you the volume of the disk at a certain time. How would you compute its mass?

Secondly, saying "let r represent the growth rate" doesn't make any sense. The growth rate will be the derivative itself. That's what derivatives are: rates at which one variable changes with respect to another variable.

 

[JNBirDy]

Step back for a second: let's say I gave you the volume of the disk at a certain time. How would you compute its mass?

Secondly, saying "let r represent the growth rate" doesn't make any sense. The growth rate will be the derivative itself. That's what derivatives are: rates at which one variable changes with respect to another variable.

Been working on it and I think I've figured it out.

dm/dt = k*S (where k is the constant or proportionality and S is the surface area)

-> S(t) = 2Pi*(h(t)r + r^2)
-> p = m(t)/V(t)
-> V(t) = h(t)*2Pi*r^2
-> p = m(t)/h(t)*2Pi*r^2

dm(t)/dt = k[2Pi(r*m(t)/p*2Pi*r^2 + r^2]
dm(t)/dt = k[m(t)/p*r + 2Pi*r^2]

Anyways, thanks.