Spherical raindrop, mass, radius, and time

[Raquel Aduriz]

Homework Statement

Separate variables and integrate to find an expression for r(t), given r₀ at t=0

Homework Equations

M=ρ(4/3)πr³, thus V=(4/3)πr³

dM/dt=Cr³ where C is a constant

The Attempt at a Solution

∫dM=∫Cr³dt

M+constant=??

I have no idea how to integrate r because it's a function of t but we're not given the function. I don't think that integrating the left side this way will be very helpful either. Any advice would be much appreciated!

 

[SteamKing]

Homework Statement

Separate variables and integrate to find an expression for r(t), given r₀ at t=0

Homework Equations

M=ρ(4/3)πr³, thus V=(4/3)πr³

dM/dt=Cr³ where C is a constant

The Attempt at a Solution

∫dM=∫Cr³dt

M+constant=??

I have no idea how to integrate r because it's a function of t but we're not given the function. I don't think that integrating the left side this way will be very helpful either. Any advice would be much appreciated!

You want to express M in terms of r. Once you do that, then dM(r) / dt = Cr³, and you can separate the variables to get r on one side and dt on the other.

 

[Raquel Aduriz]

Ok... I guess I don't really understand how to do that. Can I do that using just the M equation, or do I need something else?

 

[SteamKing]

Ok... I guess I don't really understand how to do that. Can I do that using just the M equation, or do I need something else?

You gave an expression for the mass of the raindrop in terms of the radius right there in Section 2 of the template. It's the equation M = ...

 

[Raquel Aduriz]

Alright, so I have M=ρ(4/3)πr³. What exactly do I do with this? Take the derivative and set it equal to the other? Sorry, I'm sure I'm missing something really obvious.

 

[SteamKing]

Alright, so I have M=ρ(4/3)πr³. What exactly do I do with this? Take the derivative and set it equal to the other? Sorry, I'm sure I'm missing something really obvious.

This problem is about solving a differential equation by separation of variables. Have you studied how to do this yet?

 

[Raquel Aduriz]

Yes, but a few years ago so maybe I've forgotten something key. So I have C*r*dt=ρ*4*π*dr. Is that correct?

 

[SteamKing]

Yes, but a few years ago so maybe I've forgotten something key. So I have C*r*dt=ρ*4*π*dr. Is that correct?

Not quite.

Presumably, as the rain drop falls, it gets bigger as it collects more moisture; therefore, the radius of the drop grows as time passes.

Since the drop is spherical, M(t) = ρ(4/3)π*[r(t)]³, where M and r are written as functions of time.

You are also given the condition that the change in the mass of the drop, dM(t)/dt, at any given time is proportional to the cube of the radius of the drop, or dM(t)/dt = Cr³. At t = 0, r(t) = r(0) = r₀.

You should check your differentiation of M(t) w.r.t. time. You might have to use the chain rule here, since M is a function of r, but r is a function of t.

 

[Raquel Aduriz]

I got it! Can't believe I forgot about the chain rule. Thank you so much for your help!