How does air resistance change with weight and surface area?

[scioly]

Homework Statement

This isn't exactly homework...but I'm not sure where to post it. I was reading a textbook, and got myself in a mental predicament. I understand the reasons the book gave, but I want to *really* understand.


So my question:
"Why does air resistance decrease as the surface area to weight ratio decreases (i.e. same surface area, more weight)?"

(Specifically, I was thinking about why it takes less time for a coffee filter WITH paper clips in it to fall than it does WITHOUT any paper clips in it.)

Homework Equations

I haven't taken any physics really yet; I'm working on Electricity and Magnetism now (for fun), but that doesn't really help here...so I don't know any relevant equations.

The Attempt at a Solution

OK, so here was how I got myself in an "I understand this but I don't really understand this" state:
1) I know that a falling object is going to accelerate until it reaches its terminal velocity (that is, when the upward force of air resistance equals the downward pull of gravity)
2) I also know that, air resistance aside, gravity affects all objects equally (Galileo). BUT, when air resistance is an important factor, then different weighted objects (assuming constant weight) will fall at different speeds.
3) So the downward pull of gravity remains constant. This means that, somehow, the air resistance must decrease as the surface area-to-weight ratio decreases.
4) ...And here I got stuck.

 

[NascentOxygen]

Hi scioly! http://img96.imageshack.us/img96/5725/red5e5etimes5e5e45e5e25.gif

By decreasing the area:weight ratio, I believe they mean that you keep the same weight but
decrease the surface area! Try looking at it that way and see how you go. http://physicsforums.bernhardtmediall.netdna-cdn.com/images/icons/icon6.gif

 

[scioly]

Well if you decrease the surface area, then yeah, the number of air molecules that hits the object will be less, and thus the air resistance will be less. But that doesn't explain a scenario like this one:

Experiment: Take a coffee filter and drop it. Then stick a paper clip in the coffee filter and drop it. Then stick 2 paper clips in the coffee filter and drop it. The surface area of the object remains constant, but the weight increases, and the time it takes to drop decreases. Why?

Or this one: Why do smaller water drops fall slower than bigger ones?

I get why there'd be more air resistance as surface area decreases, but why is there less air resistance as mass increases? And why does it only "affect" the object at low weights?

Thanks! :)

 

[scioly]

Oh, and I know I stated before that I didn't really have any physics knowledge...I'm in 9th grade, and so have not taken a formal physics class, but that being said, I've studied thermodynamics, electricity, and magnetism. I've also read through (most of) a Saxon physics book, although I was focusing on the electricity and magnetism parts. So if you want to use formulae and stuff to help me understand this, feel free to. I haven't taken calculus yet...so any equations involving that I can't do, but anything below that I should be able to understand.

 

[NascentOxygen]

The important formula here is

F = m.a

where F is the nett force acting downwards on the body.

Perhaps you can use that to explain your observations?

 

[scioly]

So as the mass increases, the force downwards increases as well?

 

[NascentOxygen]

So as the mass increases, the force downwards increases as well?

You would have to explain your thinking more precisely before I could give you any marks for that answer.

Another equation that crops up here is W = m.g

Can you explain this one?

 

[scioly]

Wait...no...I was wrong. The force downwards is constant (gravity) So we have an increase in mass, but downwards force (gravity) remains constant. So it's:

F/m = a

meaning that as the mass increases, with a constant downwards force of gravity (9.8 m/s free fall) the acceleration decreases. But somehow as the mass increases, the object's terminal velocity is also supposed to increase...

 

[scioly]

And terminal velocity is when the force upwards (air resistance) equals the force downwards (gravity) So I think it's established that the downwards force remains constant (Galileo's experiments). It has to be the upward force of air resistance then.

 

[scioly]

So F = ma and this force isn't "constant" like gravity? So then as the mass is increased, the upwards force of the air resistance will increase? No, that can't be right...I'm getting something wrong.

As F is increased, ma will increase as well in order for F to be equal to ma. Therefore, if m increases, the product of mass and acceleration will also increase, and therefore the force will increase. But I'm not sure if the acceleration of the object remains constant as the mass is increased. I would think that the opposite would be true: As the mass increases, the acceleration decreases.

 

[scioly]

OK, so here's another way I was thinking about it:

Assume air to be a fluid, and the coffee filter with or without paperclips to be a denser object moving through the fluid. As the density of the object increases (i. e. more paperclips in filter) the object moves faster. However, this effect is only seen with low weight, high surface area substances (kind of like low density...in my analogy) because with higher "density" substances, the effect of the relatively low resistance exerted by the air on the object is practically negligible, and all we're dealing with is the constant force of gravity.

So then my question becomes: "Why do denser objects move faster through air than less-dense ones?"

Or is this just a completely wrong way of looking at it?

 

[scioly]

I'm taking AP Chem this year, so I'm interested in what's happening at the molecular level too (and I should be able to understand, if you want to use those sorts of examples)

 

[NascentOxygen]

We can identify a number of different forces in this situation, each arising from different origins.

The object moves according to the nett force, this being the sum of all the forces.

 

[scioly]

OK, so W = m.g

weight = mass X gravity

Force of gravity remains constant (9.8 m/s not thinking about air resistance), so as the mass of an object (amount of "matter" in the object) increases, the weight of the object will increase as well.

 

[scioly]

Net force on falling object downwards = (Force downwards) - (Force upwards)

So:

Net force = gravity's force (not dependent on mass) - air resistance's force (dependent on mass)

I still don't get it :(

 

[CWatters]

OK, so here's another way I was thinking about it:

Assume air to be a fluid, and the coffee filter with or without paperclips to be a denser object moving through the fluid. As the density of the object increases (i. e. more paperclips in filter) the object moves faster. However, this effect is only seen with low weight, high surface area substances (kind of like low density...in my analogy) because with higher "density" substances, the effect of the relatively low resistance exerted by the air on the object is practically negligible, and all we're dealing with is the constant force of gravity.

So then my question becomes: "Why do denser objects move faster through air than less-dense ones?"

Or is this just a completely wrong way of looking at it?

You don't seem to have taken the hint that others have given about the nett force acting on the filter and investigated what the forces are. One of the forces is drag due to air resistance. I suggest you google/Wikipedia drag force and in particular look up it's relation to velocity.

Perhaps think about the forces involved when the filter falls at it's terminal velocity.

 

[NascentOxygen]

Net force on falling object downwards = (Force downwards) - (Force upwards)

Correct.

So:

Net force = gravity's force (not dependent on mass) - air resistance's force (dependent on mass)

Wrong. Try this again.

 

[scioly]

Net force = Force of gravity - Force of air resistance

and

gravity: not dependent on mass/velocity of object
air resistance: dependent on terminal velocity of object (because as terminal velocity increases, the drag forces decrease (i. e. air resistance decreases)) But why do the drag forces decrease as the velocity increases, and why does a greater mass mean a greater terminal velocity? (I mean on a molecular level...)

 

[scioly]

Wait...is the force of gravity dependent on mass? I remember this equation:

F = mg

This would seem to say that ALTHOUGH gravity is constant, the total gravitational force on an object is dependent on its mass. I don't get why, and I can't reconcile this in my mind with Galileo's experiments, but...if this is true then I think I understand why the terminal velocity would increase...the air resistance (upward force) would stay the same, but the force downwards would increase; hence, objects that weigh more would fall faster. But that's not what my book said...grr.

 

[scioly]

You don't seem to have taken the hint that others have given about the nett force acting on the filter and investigated what the forces are. One of the forces is drag due to air resistance. I suggest you google/Wikipedia drag force and in particular look up it's relation to velocity.

Perhaps think about the forces involved when the filter falls at it's terminal velocity.

Hi CWatters,

I tried googling drag force, but it didn't help me much. It had a bunch of stuff about the shape of the object and the flow around it. I found this:

F = 0.5 C ρA V2

A = Reference area in m².
C = Drag coefficient, unitless.
F = Drag force, N.
V = Velocity, m/s.
ρ = Density of fluid, kg/m³.

But I don't understand it yet.

Also, as for thinking about the forces involved...as I said before, I haven't taken a physics course yet, but I'm interested in the subject...so here goes:

Downwards forces:
1) Gravity: F=mg. As mass increases, gravitational force increases. On Earth, g is 9.8 m/s (I think)
2) ?? I can't think of any others

Upwards forces:
1) Air resistance (sheer number of molecules hitting against surface area): dependent upon surface area
2) Drag force: dependent on mass?? I don't get it yet...

Thanks for your help; I know it's probably quite frustrating working with me.

 

[NascentOxygen]

Net force = Force of gravity - Force of air resistance

and

gravity: not dependent on mass/velocity of object
air resistance: dependent on terminal velocity of object (because as terminal velocity increases, the drag forces decrease (i. e. air resistance decreases)) But why do the drag forces decrease as the velocity increases, and why does a greater mass mean a greater terminal velocity? (I mean on a molecular level...)

You are confusing terms. I haven't mentioned terminal velocity. I have kept discussion on FORCES.

As velocity increases, drag increases.

 

[scioly]

Oh...that's opposite from what I thought.

Wikipedia says:
"In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction) refers to forces acting opposite to the relative motion of any object moving with respect to a surrounding fluid. This can exist between two fluid layers (or surfaces) or a fluid and a solid surface. Unlike other resistive forces, such as dry friction, which are nearly independent of velocity, drag forces depend on velocity."

So is drag the same thing as air resistance? And why, as velocity increases, does the drag/air resistance increase? I still don't get it.

Also, if I say that as velocity increases, drag increases, then as the mass of a light object increases, it falls faster (still not sure why), its velocity increases, and its drag increases...and it falls slower...

(by the way, I thought that the terminal velocity was all about forces, as it's the 'equilibrium' velocity, if you will, of a falling object. It's the velocity to which an object accelerates, where the FORCE of air resistance upwards equals the FORCE of gravity...or so I thought :) )

 

[NascentOxygen]

Drag is air resistance when movement is through air. To move through any fluid what you are really doing is moving a quantity of that fluid from in front of you and filling in a gap you're creating behind you. Picture a large boat traveling in water, it is basically pumping water from in front of it to make a space in the shape of its hull for it to slide into, and pumping this water around its hull to fill in a void (also in the shape of its hull) behind it that it has moved out of. Of course, this all happens incrementally. So the faster you want the boat to move, the more water you have to pump per second. Pumping water takes effort, pumping it faster takes even more effort.

Fluid friction (aka drag) increases with speed, and with density, and with viscosity.

Your understanding of terminal velocity is correct.

I think you are getting close to understanding why larger-shaped bodies of equal weight can fall more slowly. Some soap bubbles don't fall at all.

 

[scioly]

Larger shaped bodies of equal weight would be those bodies that have more surface area; therefore, they'd fall slower because more air particles are bouncing of them (more air resistance/drag)

And when the weight is increased...

F = m * a

The acceleration due to gravity remains constant, but the mass increases, so the total force downwards increases. Therefore, since there is more force exerted on the air particles below, the momentum of the object (and the surrounding air particles) increases, and the terminal velocity increases as well.

Do I have it now?

 

[scioly]

This is the response that I got from a friend (edited a little):

Let's say that we have two masses. One is 1 kg, and the other is 2 kg. Assume that they have the same surface area, and surface, so the air resistance is the same.

F = ma.

For the 1 kg object the force is 9.81 N, and for the 2 kg object the force is 19.62 N.

9.81 N = (1 kg)(a)
a = 9.81 m/s^2

19.62 N = (2 kg)(a)
a = 9.81 m/s^2

Thus, the acceleration is the same.

Now say that we wait until 1 second after the fall. The downward speed is 9.81 m/s. The momentum for the 1 kg object is 9.81 kg*m*s^-1, and the momentum for the 2 kg object is 19.62 kg*m*s^-1.

Instead of thinking of air resistance as friction, think of it as air particles bouncing off of the objects. Assume that there is ½ mol of air particles underneath each mass, and that all of the particles have the same mass, say 32 amu, or 5.3 x 10^-26 kg per particle. For all of the particles this means a total mass of 0.032 kg. Now let's imagine that this is a solid block underneath the falling mass.

Conservation of momentum says that after the collision of two objects the total momentum of the system remains the same.

After the mass hits the air, the air will be going at the same velocity as the falling mass.

So, for the 1 kg object:

9.81 = (x)(1) + (x)(0.032)

Where x is the ending velocity of the system.

x = 9.51 m/s

For the 2 kg object:

19.62 = (x)(2) + (x)(0.032)

x = 9.66 m/s

As it can be seen, the 2 kg object is going faster after the collision then the 1 kg object. So, air resistance slows the 2 kg object less than it does the 1 kg object.

(Momentum equals velocity times mass)

So as the weight of the object increases, the air resistance doesn't decrease; rather, the effect of air resistance on the object decreases.

 

[NascentOxygen]

Larger shaped bodies of equal weight would be those bodies that have more surface area; therefore, they'd fall slower because more air particles are bouncing of them (more air resistance/drag)

And when the weight is increased...

F = m * a

The acceleration due to gravity remains constant, but the mass increases, so the total force downwards increases.

I wrote it as W = m.g
This is the same equation, but emphasises gravity being a 'constant'.

 

[NascentOxygen]

Your friend's analysis is imaginative, to say the least! I'm left with an image of the object arriving at ground level with a huge ball of congealed air stuck onto the front of it. Wouldn't that mean that the farther it fell, the slower it would become? If it fell from a great enough height, it might eventually become almost motionless, just hanging in midair after having given up most of its "momentum"? What about the work done by gravity all the while the body is falling?

Motion in a fluid involves pushing particles aside, and forcing them to follow a path around the body to fill in behind.

 

[scioly]

Hmm...not really.
1) We're assuming here that this 1/2 mol block of air is the ONLY air in the system.
2) The point is how this extra force translates into more momentum, which then translates into the object being able to travel faster, and how the only reason it has a greater velocity is because it's decreasing the effect of the air resistance.
3) The object isn't giving up most of its momentum, it's only giving a small fraction of it. Nota Bene: She's only in 7th grade...and her explanations make sense to me...the point was that we were "freezing" the air molecules, and seeing how the force exerted on them differed based on the mass...it was a thought experiment...