Gravity Near the Surface of the Earth
So far, we have learned that, near the surface of the earth, gravity causes an object to accelerate downward at a constant acceleration of g = 9.81 (m/sec)/sec.
Well, it turns out gravity causes all objects to accelerate downward at a constant 9.81 (m/sec)/sec near the surface of the earth regardless of size, shape, or mass.
At this point, you might be saying to yourself, this can't possibly be correct. You might have even come up with examples you have encountered that prove this statement false.
For instance, if gravity pulls all objects downward at the same constant acceleration of g, then when you drop any two objects, they should hit the floor at the same time if you drop them simultaneously. This is because they experience the same acceleration. As a result, they pick up speed at the same rate, thus hitting the floor at the same time.
If this is true, a feather and a coin should both hit the floor at the same time if you drop them simultaneously. You can try this at home. If you don't have a feather handy, you can use a sheet of paper. Remember if gravity pulls everything down at the same constant acceleration of g, any two objects should hit the floor at the same time if they are dropped simultaneously.
Now, if you did this, you will notice that the paper or feather (whichever one you used) takes longer to hit the floor than the coin. In other words, it stays in the air longer than the coin. So, I guess this means gravity doesn't cause all objects to accelerate at the same rate. Hmmmm........
Well, I'm going to contend that what we've stated above is still true. Near the surface of the earth, gravity causes all objects to accelerate downward at a constant acceleration of 9.81 (m/sec)/sec. At this point, you might be thinking I'm pigheaded and stubborn.
Before you make your final decision about that, ask yourself this. Is gravity the only force acting on the feather and coin when you dropped them? Because if gravity isn't the only force acting on those objects, then that could account for the fact that they didn't hit the floor at the same time. Think about this before you move on.
Well, it turns out gravity isn't the only force acting on the feather and coin. The other force acting on the feather and coin comes from the medium the feather and coin are falling through. As the feather and coin are falling, they are passing through air and running into a lot of air molecules. As they push through the air, the molecules also push back on them. Remember, Newton's 3rd Law of Motion. This pushing back by the air molecules as the feather and coin fall through the air is called air resistance. So, as the feather and coin fall, there are actually two forces acting on them. Both gravity and air resistance are acting on them. Gravity is pulling downward on the feather and coin causing them to fall. Air resistance is pushing upward on the feather and coin, keeping them in the air longer than if gravity was the only force acting on them.
This is precisely what accounts for why the feather and the coin hit the floor at different times. If there was some way we could get rid of air resistance, both the feather and coin would hit the floor at the same time. In fact, if you were to suck all the air out of a room and then tried the experiment again, you would notice that the feather and coin do, in fact, hit the floor at the same time. If you take a physics course in the future, your professor will probably show you this demonstration in class, except he won't really suck the air out of the room. The standard demonstration of this is with a feather and coin in a sealed clear tube. With the air still in the tube, the feather will take longer to hit the bottom of the tube. However, after the professor sucks the air out of the tube, both the feather and the coin will hit the bottom of the tube at the same time. In the following picture, the tube on the left has air in it. In that tube, the coin will hit the bottom of the tube first because of air resistance. In the tube on the right, the air has been sucked out. As a result, both the coin and the feather will hit the bottom of the tube at the same time because there is no air resistance in the tube. (Let me apologize for my insensitive portrayal of a feather.)
If you are having difficulty thinking of air resistance and how it affects falling objects, you can think of the air as water. If you dropped an object in a pool, the object would encounter resistance as it falls because it has to push through the water. (The resistance is coming from the water pushing back on the object as it falls through the water. Once again, due to Newton's 3rd Law of Motion.) Likewise, if you dropped two objects in a pool, you would not necessarily assume both would hit the bottom of the pool at the same time because the two objects might encounter different amounts of water resistance as they fall, depending on their shape.
In conclusion, near the surface of the earth, gravity causes all objects to accelerate downward at the same constant acceleration of 9.81 (m/sec)/sec. The only reason we don't see this all the time is because of the pesky air resistance.
Now that we have talked about the acceleration that gravity causes objects to experience, what about the force? How do we measure the force of gravity on objects near the surface of the earth?
Well, this is not too hard of a matter. First, recall Newton's 2nd Law of Motion. If you will recall, Newton's 2nd Law of Motion tells us how an object responds to a force. In other words, it tells us how an object accelerates in response to a force because the acceleration an object experiences is a direct result of the force applied to the object. The formula is as follows.
a = F/m where a is the acceleration, F is the force, and m is the mass.
Next, let's rewrite this formula by multiplying both sides of the equation by m, the mass. The result is as follows.
F = ma
Okay, now we have the formula we need. F = ma. To figure out the force with which gravity pulls an object downward (near the surface of the earth), all we have to do is use this formula.
We already know the acceleration an object experiences in response to gravity. It is 9.81 (m/sec)/sec downward. However, let us write it as g to save us some space.
Therefore, we know a = g in the formula above. By putting this into the above formula, we get the following result.
Fg = mg
where Fg = the force that gravity exerts on an object of mass, m
(The "g" in the subscript above just denotes that we are talking about the force of gravity.)
Well, that's fairly easy. To figure out the force that gravity exerts on an object, all we have to do is multiply the mass of the object (in kilograms) by g which is 9.81 (m/sec)/sec. Since we are using S.I. units, the answer you get should be in units of Newtons, which is the unit for force. Refer to this section if you need to review units.
Let's do some examples. In both examples below, assume that the objects are near the surface of the earth.
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Example 1:
What is the force that gravity exerts on an object that has a mass of 5 kilograms?
Looking at the formula above, all we have to do is multiply the mass of 5 kilograms by g which is 9.81 (m/sec)/sec.
The answer is that the force is 49.05 N downward. As expected, it is downward because gravity pulls things down.
- Example 2:
We know that gravity pulls down a 5 kg object at the same acceleration as a 3 kg object. However, are the forces the same?
Well, obviously the answer is no. While the acceleration might be the same, the force of gravity is different because the force that gravity exerts on an object depends on its mass. If two objects have different masses, then the force that gravity exerts on them is different. You can see this explicitly by looking at the formula above and calculating the force that gravity exerts on a 3 kg object and a 5 kg object. We will talk more about this in a later section.
Well, you might be wondering why we bothered talking about the force of gravity if we already knew that it accelerates objects downward at a constant 9.81 (m/sec)/sec. It turns out that the force of gravity is a little more relevant than you think. In fact, some people are obsessed with the force that gravity is pulling on them.
In fact, the force that gravity exerts on an object (near the earth's surface) is what we commonly refer to as its weight.
In other words, W = Fg where W is the weight of the object.
In terms of the formula above, W = mg where W is the weight of an object of mass, m.
(This is the form of the formula most people are familiar with.)
This shouldn't come as too much of a surprise because the scale you use to measure your weight is precisely measuring how hard the force of gravity is pulling you downward. In fact, "pounds" is a unit for force much like the "Newton". It is an English unit and not a metric unit, but we can convert from one unit to the other.
Specifically, the scale is measuring how hard you are pressing down on it. However, what is causing you to press down on the scale (when you are standing on it) is the force of gravity pulling you down. Therefore, the scale is measuring how hard the force of gravity is pulling down on you. This is what we commonly refer to as weight.
Finally, if you were observant, you might have noticed that, in the situations we discussed above, I was careful to say that the acceleration due to gravity was a constant 9.81 (m/sec)/sec downward only near the surface of the earth. That's because the acceleration due to gravity isn't constant in reality, but it changes by such a small amount near the surface of the earth that the assumption of a constant acceleration still gives us a reasonably accurate answer. I will get to this point in the next section when we discuss about gravity in a little more detail.
