The Physics Philes, lesson 6: Nothin’ But Net Force
In which the net force is measured, a crappy drawing is produced, and a question is asked.
The bell has rung (ringed?) girls and boys. Time to settle down and let Mindy teach you a little physics. Also, you need to listen so you can tell me everything I get wrong. Everybody wins!
If you’ll remember, last week we started down the road of Newton’s laws of motion. The first law of motion applies when the net force acting on a body is zero. This week we’ll discuss Newton’s second law of motion.
The second law of motion applies when the net force acting on a body is not zero. (Remember, a net force is the sum of all the forces acting on a body.) A net force acting on a body causes the body to accelerate in the same direction as the net force. This also applies when the body is moving along a curved path. There isn’t a lot of math here. It’s been proven by experiments. That evidence shows that for any given body, the magnitude of the of the acceleration is directly proportional to the magnitude of the net force acting on the body. And for a given body, the ratio of the magnitude of the net force to the magnitude of the acceleration is constant, regardless of the magnitude of the net force.
What does that mean in real life terms? Let’s use that classic physics example: the hockey puck. If you push a hockey puck across the ice at force x, it will accelerate a. But if we exert force 2x on the puck, the acceleration will increase in the same proportion. Get it? Not too hard.
Anyhoo, this ratio is called the body’s inertial mass, or just mass. Mass is a quantitative measure of inertia. But what is inertia? Inertia states that the greater the mass of an object, the more the body resists being accelerated. The relationship between mass, force, and acceleration is expressed in newtons. One newton is the amount of net force that gives an acceleration of one meter per second squared to a body with a mass of one kilogram. Or, in math-ese:
1 N = 1 kg • m/s2
Are you still with me? Good. Remember last week when we discussed the superposition of forces? You remember. When a combination of forces are applied to a body, the body will move in the same direction and have the same magnitude as a single force equal to the vector sum of the applied forces. Well that principle applies even when the net force is not zero.
The conclusions I’ve just attempted to explain are important, because Newton took them and wrapped them up in a pretty bow to create the second law of motion.
You see, this law is the basic relationship between force and motion. Stated in words, it says:
If you think about your day-to-day, it makes sense. If a net external force acts on a body, the body accelerates. The direction of the acceleration is the same as the direction of the net force. The mass of the body times the acceleration of the body equals the net force vector. Or, in math:
ΣF = ma
a = ΣF / m
(Note: From now on I’m just going to bold vectors. I can’t be bothered to figure out how to put an arrow over the top.)
But! This is a vector equation! It’s much more helpful if we use its component form:
ΣFx = max
ΣFy = may
ΣFz = maz
A couple of important things to keep in mind when working with the second law of motion. The second law refers to external forces, i.e. forces exerted on the object by the environment. In addition, this law only applies when the mass of the body is constant. If the mass isn’t constant, I’m told by my trusty textbook that momentum is a more useful concept. (I have no idea. I guess we’ll find out when I hit that section, eh?) Finally, Newton’s second law of motion is valid only in inertial frames of reference. An inertial frame of reference is a frame of reference that is at rest or moving at a constant speed. For our purposes, the Earth is an inertial frame of reference.
Before I move on to an example, I have a question about inertial frames of reference that I hope you can help me with. How do we know what is an appropriate frame of reference? Is it always the Earth, as long as the problem exists there? Could it ever be anything else? Please. Halp!
OK. Let’s do a sample problem.
This problem deals with determining acceleration from a given force. A person applies a constant horizontal force with a magnitude of 20 N to a box with a mass of 40 kg. The box is sitting on a floor with negligible friction. What is the acceleration?
The first thing we have to do when working with forces is choose a coordinate system and identify the forces.
I mapped out three forces: the force exerted on the box (20 N), the normal force (n), and the weight (w). I chose to make the x-axis parallel to the floor. The problem says that friction is negligible, so we don’t have to put it in our diagram. There is also no vertical motion, so ay = 0. All we have to do is solve the equation for a.
∑Fx = max
20 N = 40 kg(ax)
Remember, a newton can also be written kg * m/s2. All we have to do now is just work it through.
20 kg • m/s2 = 40 kg(ax)
20 kg • m/s2 / 40 kg = 40 kg(ax)/40 kg
.50 m/s2 = ax
Just divide the force by the mass, and you get the acceleration! Easy peasy lemon squeezy. That’s a pretty simple problem, but you get the idea.
Now I throw it to you. Did I miss anything? Did I gloss over something important? I’m still learning. Don’t be afraid to school me on the finer points of Newton’s second law of motion. And don’t forget to tell me about inertial frame of reference. Please!
Featured image credit: kennymatic