# 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/s^{2}

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** = m**a**

**a** = **Σ**F / 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:

ΣF_{x} = ma_{x}

ΣF_{y} = ma_{y}

ΣF_{z} = ma_{z}

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 a_{y} = 0. All we have to do is solve the equation for a.

∑F_{x} = ma_{x}

20 N = 40 kg(a_{x})

Remember, a newton can also be written kg * m/s^{2}. All we have to do now is just work it through.

20 kg • m/s^{2} = 40 kg(a_{x})

20 kg • m/s^{2} / 40 kg = 40 kg(a_{x})/40 kg

**.50 m/s ^{2} = a_{x}**

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*

## 5 Comments

Well, if mass is changing, I guess one should use the extended Newton’s second law (the original version):

F=m*a+(mass change over time)*v.

This one requires calculus to understand why, though. One situation that would require for this is the rocket. Rockets are continually losing mass, after all.

For calculus people:

F=dp/dt=d(mv)/dt

Use chain rule:

F=v*dm/dt+m*dv/dt

dv/dt=a

Also, another cool thing about Newton’s second law is that this law can describe any kinds of motion as long as you set up the right equations. It can describe pendulum swings, spring bouncing back and forth, circular motions, free fall, waving of strings, elliptical shape of planetary orbits, terminal velocity from drag, and more.

But, sometimes one feels lazy in solving Newton’s second law or the kinematic equations. There is another way to describe motion of objects which can be easier depending on the situation. I am sure it will come up in later editions of this post.

An appropriate frame of reference is a frame that is not accelerating. For example, when a car is speeding up, and you feel your body pressing against the back of the seat, the car is not appropriate. Meaning, if you shut all the window, and you don’t feel anything moving, it becomes an appropriate frame of reference. It can be that the place you are trapped in is moving at constant velocity or standing still, you don’t know, but it doesn’t matter.

Also, a car that is turning a corner, even if it keeps moving at the same speed, is NOT an appropriate frame. In this case, velocity is changing because the direction is changing, which means there is an acceleration changing that velocity, which means you will feel your body pressing against the door or the passanger unfortunate enough to be sitting near you. So yeah, rotating frames are not good frame of references.

Now, you might be thinking: Hey! The Earth is rotating! Yeah, but the Earth is so huge that the turn of the Earth is not really sharp. It’s effect is small enough that you can ignore it (physicists do a lot of ignoring, it prevents problems from getting out of hand). Of course, at huge scales, you can’t ignore it, since things like the coriolis effect affects hurricanes. But this is the human scale, and it is freaking small.

A frame of reference, by the way, can be thought of as an imaginary grid moving at constant or zero velocity, but it is much more convenient to fix the 0 point on an object at time=0. So, you throw a ball up and down in a car moving at say 50 mph going straight. So set up a grid that goes at the same velocity as the car. From the perspective of the person in that frame, the ball is just going up and down, not moving forward at all. From the person standing still, the ball follows a curved path, and the person inside it moves forward at the same speed. The person and the ball, though, remains in the same position relative to the moving grid. In that grid, the person is standing still and the ball is just moving up and down. See now that solving physics problem outside that grid is the same as solving physics problem inside the grid?

But what if the car started to speed up? Then the ball, as it goes up, start moving backwards faster and faster, and hit your face! But that means that Newton’s second law is screwed up in an accelerating frame. The ball started accelerating backwards even though no force was acted upon it! What happened, of course is that the car was accelerating, but the stuff inside it not attached to the car kept trying to move in the same velocity (Newton’s first law). The car was overtaking the ball because of that. Now you see why Newton’s second law fails in an accelerating frame. You also see now that solving physics problem outside that grid is now not the same as solving physics problem inside that grid. Using non reference frame is just a pain the butt. Let’s not use it, please.

Shorter answer: Yes, it can be anything and anywhere. Also, Newton’s first law.