Basics: Work Energy

**Pre Reqs:** [What is a Force](

[Previously, I talked about the momentum principle]( Very useful and very fundamental idea. The other big (and useful) idea in introductory physics is the work-energy theorem. Really, with work-energy and momentum principle, you will be like a Jedi with a lightsaber and The Force – extremely powerful.

Well, what is work? What is energy? How are they related? In [another post, I talked about energy.]( I think it is interesting to look at how most textbooks define energy:

*Energy is the ability to do work*

This is really a stupid definition. Kind of circular logic, if you ask me. In the post I mentioned earlier, I claim there are two kinds of energy, particle energy and field energy. At low speeds (not near the speed of light), particle energy can be written as:

![Screenshot 53](

Where *m* is the mass of the particle, *c* is the speed of light. So, if you just look at a particle, that is it for the energy. Now, what about the “work” portion? Work is defined as:

![Screenshot 54](

Where *F* is the net force on the particle, ?r is the vector displacement of the particle. The “dot” in between F and ?r represents the “dot product” operation between vectors (also known as the scalar product). In a [previous post]( I showed that you could multiply a scalar quantity by a vector quantity. Here I need to do “something” with two vectors. You can’t multiply two vectors in the same sense that you multiply scalars. A general definition of the dot product for two vectors:

![Screenshot 55](

That looks a little more messy than I wanted, but it can not be helped. Really, it is not that complicated. The dot product is simply the projection of one vector on the other. Let me explain in terms of work.

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Spring Motion and Numerical Calculations

Maybe you know I like numerical calculations, well I do. I think they are swell. [VPython]( is my tool of choice. In the post [Basics: Numerical Calculations]( I used vpython and excel to do something simple. I will do that again today (in that this problem could also be solved analytically). However, there is one big difference. This problem has a non-constant forces. Suppose I have a mass that is connected by a spring to a wall. This mass-spring is sitting on a table with no friction.

![Screenshot 27](

There is a very interesting property of springs. The more you stretch them, the greater the force they exert (in the usual model of springs). This model works very well.

![Screenshot 28](

This is known as Hooke’s law. I have written it as a scalar for simplicity. The “k” is called the spring constant. It is a measure of how “stiff” the spring is. The value “s” is the amount the spring is stretched. Typically, there is a minus sign in front of the ks to indicate that the force is in the opposite direction that the spring is stretched. Really, in a scalar equation this is rather silly to include (but everyone does anyway).

**Question: What will the motion of the mass be like if I pull it back and then let go?**

Although this can be determined analytically, I am going to first calculate this with vpython. I will try to show all the details so that you can reproduce this also. If you have not already installed [vpython](, do that now (don’t cost nothing).

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Basics: Numerical Calculations

**Pre Reqs:** [Kinematics](, [Momentum Principle](

What are “numerical calculations”? Why are they in the “basics”? I will give you really brief answer and then a more detailed answer. Numerical calculations (also called many other things – like computational physics) takes a problem and breaks into a WHOLE bunch of smaller easier problems. This is great for computers ([or a whole bunch of 8th graders]( because computers don’t mind doing lots of little problems. Why are they “basic”? Well, most text would say they are not basic. I disagree. I think this is a legitimate method for solving problems. In particular, this is a great way of solving problems that can not be solved analytically (meaning solving one hard problem).

**Numerical Calculations are Theoretical Calculations**

Let me just get this out of the way. Numerical calculations and analytical calculations are really in the same “class”. Often people will lump numerical in with “computational experiment” but that is a really bad thing to do. Some others will claim that there are three different “paths” to discover stuff in science: theory, experiment, and simulations. Simulations are the same thing as numerical calculations which are the same as theory. ([I wrote a letter about this in the American Journal of Physics](

**Example Problem**

Let me start with a problem that can be solved analytically. Suppose I have a ball of mass 0.5 kg and I throw this straight up with a speed of 10 m/s. How high will it go?

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Basics: Forces and the momentum principle

**Pre reqs:** [Free Body Diagrams](, [Force](, [Kinematics](

The time has come to look at things that are NOT in equilibrium. The most basic question to ask yourself is: *”What do forces do to an object”*? Aristotle would say that forces make things move. Constant forces make things move constantly. Actually, Aristotle said there were two types of motion:

  • Natural motions: These motions don’t need anything to happen, they just do. Example: a rock falling. You don’t need to do anything to it. Example: fire rising. It just rises. (there was more to it than that, but you get the idea).
  • Violent motions: These motions are due to some interaction that forces them from their natural state. The natural state of a cart is to be at rest. If someone pushes on it, it will move. When you stop pushing (stop the violent motion) it returns to its natural state – at rest

I am talking about Aristotle, because these basic ideas are what most people think. If you push something it moves. If you stop pushing, it stops. And these people are correct. The problem is that there is always this extra force that no one thinks about – friction. Without friction, the rules change.

**New Rules (Newtonian ideas)**

If you push something with one force, it changes velocity. If you stop pushing, it stays at a constant velocity.

If you want to test your feelings for force, [try this force game I made on Scratch]( The idea is that you need to move the box to the red circle. The arrow keys exert a **force** on the object.

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Basics: Friction

**Pre reqs:** [Free body diagrams](

Friction is an interaction between two objects in contact that opposes relative motion of those two objects. It is not something fundamental (like gravity, or electromagnetic force), but it comes up enough that it will be worthwhile to talk about it. Let me start with a simple example. Suppose I have a book on a table. Here is the free body diagram for the book:

![Screenshot 27](

Simple enough – right? There are two forces on the book. A contact force (the table pushing up) and a long range force (the gravitational force of the Earth pulling down on the book). These two forces have the same magnitude, so when added together, they give a total of zero vector. This means the book is in equilibrium.

Now, what if I push on the book from the side? Suppose I push with 1 Newton. If the book is still in equilibrium, what does that mean? It means the free body diagram must look like this:

![Screenshot 28](

If the book is still in equilibrium, then the force of the table on the book (due to friction) would have to have the same magnitude as me pushing on the book. Note: Even though my push and friction are equal and opposite, these are not Newton’s third law force pairs – I talked about that in the free-body-diagram post.

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Basics: Free Body Diagrams

**Pre Reqs:** [Intro to Forces](, [Vectors](

Hopefully now you have an idea of what a force is and what it isn’t. What do you do with them? The useful thing to do with forces is to determine the total force acting on an object. At the beginning of the introductory physics course, you will likely look at cases where the total force is the zero vector. This is called equilibrium. Even if you are looking at cases where the forces don’t add up to the zero vector (I say that instead of just “zero” to remind you that the total force is still a vector). Physicists like to represent forces on an object by drawing a Free Body Diagram. This is simply a representation of an object and a graphical representation of all the forces acting on that object.

Simply put, in a free body diagram, all the forces acting on the given object are represented as arrows. Let me start with a simple case, a box sitting on a table.

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Basics: What is a Force?

**Pre-reqs:** None.

I intend to talk about forces and force diagrams, but there is a more fundamental question to address first. What is a force? Most texts define it as a push or a pull. That really isn’t a bad definition. Maybe a better (or maybe worse) definition would be “forces are things that change the motion of an object” (change being the key word). If I had to choose one definition of force, it would be something like this:

**Force:** *A force is an interaction between two objects. There are 4 known forces:*

  • Gravitational force: An attractive long range force between objects with mass
  • Electromagnetic force: An attractive or repulsive long range force between two objects with charge
  • Strong Nuclear force: An attractive short range force between particles like protons and neutrons
  • Weak Nuclear force: A short range force responsible for beta decay. *Yes, I know that is a confusing force – for introductory physics, you won’t use this force*

All forces are some form of the above forces.

**Important properties of forces**

  • Forces are an interaction between TWO objects. It is not possible to have a force on an object and not have another object involved.
  • Forces are vectors. They have magnitude and direction
  • The unit for force is the Newton. If you do a whole bunch of cool stuff, they will name something after you also.
  • Forces are NOT properties of an object like mass or speed or color. They are properties of an interaction between two objects. Yes, I already said that, but it is important.

There are some more things about force you will need to know. For now, this should be enough.

Basics: Relative Velocity

**pre reqs:** [Vectors and Vector Addition](

This was sent in as a request. I try to please, so here it is. The topic is something that comes up in introductory physics – although I am not sure why. There are many more important things to worry about. Let me start with an example. Suppose you are on a train that is moving 10 m/s to the right and you throw a ball at 5 m/s to the right. How fast would someone on the ground see this ball? You can likely come up with an answer of 15 m/s – that wasn’t so hard right? But let me draw a picture of this situation:

![Screenshot 15](

The important thing is: If the velocity of the ball is 5 m/s, that is the velocity with respect to what? In the diagram, I listed the velocity of the ball as *vball-train* this indicates it is with respect to the train. There are three velocities in this example.

  • The velocity of the ball with respect to the train
  • The velocity of the train with respect to the ground
  • The velocity of the ball with respect to the ground

These three velocities are related by the following:

![relative v](

**note**: The way I always remember this is to arrange it so that the frames match up on the left side. That is to say v(a-b) + v(b-c) – you can think of this as the “b’s” canceling and giving v(a-c).

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Basics: Making graphs with kinematics stuff part II

**pre-reqs**: [kinematics]( *I don’t think you need [part I of this]( if you don’t want*

So, you still want to make a graph with that kinematics data? You think that graphs on paper are too barbaric? Well, if you are ready, you can use a spreadsheet. But be careful. If you don’t know what you are doing, you can cause some damage (much like flying a 747 after reading a blog about it). Speadsheets allow you to do a couple of things.

  • make pretty graphs
  • fit mathematical functions to data

Of course they actually do much more – but you need [“clippy”]( to help you with that.

First, what software do you use? I think most people will immediately go for Microsoft Excel. I have to admit, this is what I use because I am so familiar with it. Many people already have this also. Truthfully, it is a good spreadsheet program (but not perfect). There are some free alternatives:

  • Open Office – I use the Mac OS X variant Neo Office
  • Online spreadsheet like Zoho) or Google Docs. Both of these are fairly useable.
  • Other – like Apple’s spreadsheet or other non-free stuff.
  • A final excellent option is Vernier’s Logger Pro. Although it is not free (nor perfect) it is not too expensive and can be covered by a school site license

For this tutorial, I will show explicitly how to make graphs using MS Excel. I was going to use open office, but in order to fit a polynomial to data, you have to do some more serious stuff. The basic idea is the same no matter what you use.

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Basics: Making graphs with kinematics stuff

**pre reqs:** [kinematics](

Suppose there is some experiment in which you throw a ball up and collect position and time data (with video analysis). What do you do with this data? Your instructor told you to make a graph, but how do you do that?

Here is the fictional data you (or I) collected:


Here is the text file with the data if you want to reproduce the graphs I make here [kinematics data](

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