# Mass questions

1. Apr 12, 2005

### ConcealedDreamer

If I drop 2 of the same object but different mass like a pocket book and a encylopedia, they drop at the same time. Why is that? Anyone can get a detailed explanation? Thanks!

2. Apr 12, 2005

### dextercioby

If and only if you're doing the experiment in vacuum.Assuming that,Newton's law of gravitation attraction will give the same acceleration for two bodies with different mass falling in the earth's gravitational field.That's all to it.

Daniel.

3. Apr 12, 2005

### SpaceTiger

Staff Emeritus
It's because the force that's pulling down on your objects is the gravitational force:

$$F=\frac{GM_em}{r^2}$$

That's Newton's Law of Gravity, where G is the gravitational constant, Me is the mass of the earth, m is the mass of your pocketbook (or encyclopedia), and r is its distance from the center of the earth. What else do we know about forces? Well, Newton's Second Law of Motion says:

$$a=\frac{F}{m}$$

That is, an object's acceleration is equal to the force acting on it divided by its mass. I just said that the force acting on it was gravity, so that gives us

$$a=\frac{GM_em}{r^2m}=\frac{GM_e}{r^2}$$

This tells us that the acceleration an object experiences from gravity is equal to the gravitational constant times the mass of the earth divided by its distance from the center squared. If your objects are accelerating at the same rate, it means this number is the same for both of them, and if you take a look, you'll see why that is. The gravitational constant is universal, so it won't vary from object to object. The mass of the earth certainly doesn't depend on the object being used.

Now, you might say that the two objects have a slightly different distance from the center (and it's certainly possible that they do), but remember that we're very far from the earth's center. That is, the average distance to the earth's center from the surface is around 6,380,000 meters, while the two books you're holding up probably won't have a difference in distance from the center that's more than a few centimeters. Thus, r=Re, the radius of the earth, and their accelerations will be the same, for all intents and purposes. It's because this number is roughly constant on the earth's surface that we talk about another constant, g:

$$g=\frac{GM_e}{R_e^2} \simeq 9.8 \thinspace\thinspace m/s^2$$

This is the acceleration you'd expect your books to have.

So, you may ask, why doesn't a feather hit the ground at the same time? The answer to this question is a bit more complicated and it has to do with "air resistance". It turns out that all falling objects experience a slight upward force from the air molecules that they're falling through, but you don't notice it unless you object has a high surface area and low mass (as with a feather). It's for this reason that your books wouldn't hit the ground at exactly the same time, but the difference would be hard to measure without decent equipment.

Last edited: Apr 12, 2005
4. Apr 12, 2005

### axl

that's _not_ all there is to it, dexter.

One must postulate the equivalence principle, i.e. the gravitational mass of a body is identical to inertial mass of the same body.

5. Apr 12, 2005

### SpaceTiger

Staff Emeritus
No need to postulate anything. Measurements indicate that to the uncertainty that this person can observe, they are equivalent. Whether or not they're exactly equivalent doesn't matter for his/her purposes.

6. Apr 13, 2005

### BigStelly

IF you want evidence that the feather falling at a different speed is not due to its mass.... put the feather on top of one of your books then drop the book, this is so beautiful when you do this.

7. Apr 13, 2005

### dextercioby

YOU ARE RIGHT. I remember,sorry for the imprecision. It's the first thing taught in a classical mechanics course when introducing Newton's gravity theory...

Daniel.

8. Apr 14, 2005

### Tom Something

let's assume that the pocket book is half the mass of the encyclopedia.

are you familiar with the formula F=ma (force acting on an object is equal to the mass of the object times the acceleration caused by the force)?

here, we are interested in acceleration, so with a little division, we come to a new formula: a=F/m.

the only force (F) acting here is gravity. an object of double mass should have double gravitational pull. if you double "m", you have to double "F".

so let's say F is the force (weight) acting on the pocket book and m is the mass of the pocket book. If the encyclopedia has twice the mass, then the encyclopedia's mass is 2m. it's weight is therefore 2F.

so, the acceleration of the handbook can be represented now as
a=F/m

and the acceleration of the encyclopedia would be
a[encyclopedia]=(2F)/(2M). The 2's cancel out. That's where the magic happens.
a[encyclopedia]=F/m, just like the pocket book.

mass and weight are different. weight is how heavy something is. mass is an object's unwillingness to change. if you double the mass, you double the weight. these factors cancel each other out for downward acceleration in a vacuum.