Newton's Laws 1D:

If there is only one thing that you learn well in PH211 (and there should be several things) then Newton's Laws are the most relevant for nearly all students. To allow a clear focus on Newton's Laws this first discussion will be limited to forces acting along a line in some direction. This is what we mean by 1D.

Kinematics vrs Dynamics:

Kinematics is the description of what objects do. If you can describe how the object moves in space you can (with calculus or graphing) determine it's velocity and acceleration at any point. Dynamics is the explanation for why an object moves the way it does. These two approaches are complementary and each is valuable in it's own right. There is lots of discussion in the physics community about whether to start with kinematics or dynamics. We are going to start with the dynamics and extract it's relationship to the kinematics. This connection will use tools that you're only just learning in MTH 252 but hopefully will provide some added insight for your calc class.

What System?

A common challenge in the early stages of learning to work with physics concepts is not being clear about what object or group of objects is the focus of your analysis. As you get more experienced over the years you will find that you may define different systems within the same problem depending on your immediate needs. Here is an effective working definition of the system from my colleagues at ISLE (Eugenia Etkina et al):

" A system is the object we choose to analyze. Everything outside that system is called its environment and consists of objects that might interact with the system (touch, push, or pull it) and affect its motion through external interaction."

By being explicit about what we think of as the system we can more clearly identify external interactions with the object of interest. Mechanically we indicate the system by sketching a line (often dashed) around whatever we're are calling the system.

Thinking about the first step in our problem solving process this means that on your initial sketch for any problem you will draw some sort of bubble around the object or objects that are the focus of your analysis.

Velocity and Speed:

As we explore and discuss Newton's Laws in 1D it will not be particularly clear how velocity and speed are different. In some contexts people use them interchangably though they do represent different concepts. At this point just learn to notice the difference in the terms and when we get to our discussion of vectors in another week or so the difference will be explored more explicitly.

Velocity (speed) and Acceleration:

The need to distinguish clearly between velocity and acceleration will come up routinely in our explorations of Newton's Laws. We will talk about this again next week in kinematics but for now we need to get clear about the core difference between the two. In general conversation in the world you should assume that most folks use the terms acceleration and velocity or speed pretty much interchangeably. When speaking to another STEAM person we can hope that their meaning is clear but it is still worth listening carefully to be sure what they mean.

If we take a calculus approach to these terms you may remember that velocity is the slope of the position function (in 1D) and acceleration is the slope of the velocity function. In the notation of calculus we would write these as...

v = dx/dt and a = dv/dt

In math class this is a notation that indicates something different than a mere fraction. None the less, in applied sciences, we will think of dx as just a small version of Δx which is the short hand for (x_later - x_earlier). We are making the point the Δx isn't always associated with initial and final but more generally it is just a difference. We can then write...

v = Δx/Δt and a = Δv/Δt

If you reflect carefully on this math you will note that Δx/Δt describes the slope of the straight line that spans Δx. This means that Δx/Δt is the average (constant) speed that would get you between the two positions in the same amount of time.

v_average = v_ave = Δx/Δt and a_average = a_ave = Δv/Δt

 

v_{instantaneous} = v = dx/dt and a_{instantaneous}= a = dv/dt

 

As you will remember from calculus acceleration is the rate of change of velocity often written a = dv/dt. At it's core this means that if dv or Δv = 0 (no change in velocity) then a = 0. Alternatively, if a is NOT equal to 0 then the velocity must be changing. In one (1) dimension this matches our intuitive expectation that this means the speed is changing. Be careful --- it's not this simple in more than one dimension!

Take Away: The important take away here is that velocity (speed) describes the way in which an object is changing position and acceleration describes the way in which the velocity is changing. You can have one without the other in any sense or both at the same time.

Activity: Identify physical circumstances when each of the following are true:

i: An object is moving but has no acceleration (a = 0). Describe how the movement of the object evolves.

ii: An object is acceleration but is not moving (v = 0). Describe how the movement of the object evolves.

iii: An object is moving AND accelerating. Can such an object be slowing down?...speeding up?....neither?

You will hopefully note from the definition of acceleration that both slowing down and speeding up represent acceleration because the speed is changing. In physics there is no 'deacceleration'.

Forces:

One of the ways in which f2f classes are interesting is that your teacher can manage your access to information for the purposes of generating learning opportunities. This is not really possible in this sort of an asynchronus learning tool. To get the most out of this learning opportunity try to explore any activities as you get to them without looking too far ahead for 'the answer'.

Activity: Consider: What is a force? How do you recognize one in the wild? A good way to do this is to make a list of things that you think are forces and look for commonalities among items on your list. We will probably approach this in class via a real or virtual whiteboard to build a list and discuss it.

Just so you don't feel alone wondering why this is, or isn't, a confusing question here is an insightful Veritasium video (yes, Derek is awesome!).

Take Away: Force is the name we give to any interaction between two objects that can be characterized as either a 'push' or a 'pull'. All forces that we know of fall into one of two categories. There are forces which require a physical contact between the two objects (called contact forces) and most forces fall into this category. There are a few forces (4 actually) that do not seem to require physical contact (called non-contact forces - brilliant!). Gravity is the only non-contact force we will deal with this term.

Freebody Diagrams (variously called Force Pictures or Force Diagrams)

Given that every physics and engineering class depends on freebody diagrams (FBD's) you might expect there would be a consistent and universal way that they are taught across the world. It is slightly startling to me that there is so much variation in how we approach this tremendously important topic. Here is my perspective on this foundational tool for analysis.

Below are the features of the first section of our problem solving framework. You will, perhaps, note that they are primarily dedicated to clearly articulating and communicating the setting and characteristics of the problem. For any problem where forces play a role the freebody diagram is a simplified representation of the forces that you have indicated on your sketch. On your sketch you have indicated the forces you believe play a role in the problem including their directions. Whether you are correct or not is less important than knowing how to follow the logical and mathematical consequences of the forces you believe are present. If you discover you forgot a force then you can just go back and add the force to your sketch and move forward to determine the consequences.

Problem Solving Framework:

A: Framing:

Visual Representation: a basic sketch of what is happening, the important quantities (forces, velocities, masses, energy, etc) and what you are solving for.

Relevant Concepts: possible physics concepts related to setting

Similar Problems (this is an internal discussion): past problems that may be relevant

Assumptions and Simplifications: Details you are explicitly neglecting

Information Needed: Values stated in the problem and other values you might need to look up or estimate

Sample Problem:

Consider an elevator moving upwards towards the 3rd floor. A pygmy giraffe is standing in the elevator and a large fruit bat is hanging upside down from a rope tied to the ceiling of the elevator.

Steps to a successful Freebody Diagram (FBD):

1: Sketch the situation described in the problem.

Consider what you might have sketched but note that there is clearly no requirement for artistic skill!!

 

2: Circle an object or objects of interest in the sketch — this the system.

In this setting I have circled two possible objects of interest - the giraffe like creature circled with a dotted line and the possibly bat like creature circled with a dashed line. We will start by focusing on the fruit bat. I am now seeking to determine and indicate any and all forces that I think are acting ON(!) the bat. This is an important reason for defining the system clearly so that we don't get distracted.

3a: Forces: Non-Contact (gravity)

While it may seem like we're overthinking things we will eventually meet problems where gravity may not be present or may be negligable. Either way it takes just a moment to decide whether gravity is a player in the problem. Draw an arrow on the sketch indicating direction of the force. We generally indicate the force of gravity using the symbol F_g.

3b: Forces: Contact (everything else)

Because all other forces must be contact forces we now look at our sketch and seek to identify what is touching the object in my system. For each contact I must decide to either indicate the direction of that force OR decide to neglect it as not relevant to the problem. In this case I note that the bat is 'touched' by the rope he is holding on to and by the air in the elevator. The force ON the bat is upwards and I will label it as F_rope. I am going to neglect the contact force from the air because I don't know how to deal with it at this point AND it seems to me that the behavior of the bat is pretty much the same whether there is air in the elevator or not (aside from the breathing problem). Now my sketch looks like this:

4: Simplify:

We now simplify the object of our interest to a point. In the long run we will need to consider both the overall motion of the object and it's orientation in space (rotations etc). For now we are only tracking the behavior of the object as a whole which we can represent by a point. To be most correct we would note that the point represents the center of mass of the object but we have no definition of the center if mass yet so that will wait.

5: Choose and Label Coordinate(s) Axes:

This step sets the stage for all the mathematical work we will do to solve the problem. It is important to note that you may choose any reasonable coordinate system and solve the problem. Some choices may make the mathematics more challenging but the problem can be solved. Learning to make prudent choices of coordinate systems is part of the skill set you are growing. In 1D cases it can seem a little trivial since the best choice is the obvious one -- choose your axis along the direction that all the forces align with. Call it what ever you want. Don't be trapped by classic visions of x and y directions. Explicitly indicate and define which way is the (+) direction along your choice of axis. Clearly indicate the origin of your coordinate system which will typically be the location of the point representing the object. This may sound like a lot of excess work but we are seeking to establish patterns that will be critical as problem settings grow more challenging.

6: Redraw Force(s):

Finally, redraw the forces from your sketch onto your coordinate system. Generally we draw each force as originating (the tail of the arrow) at the point which represents the object and pointing in the appropriate direction along the coordinate axis. This helps minimize potential confusions as we later translate this freebody diagram into math. At this point your simplified drawing on a well defined coordinate system with the forces added is the freebody diagram (FBD).

Note:

Note that the freebody diagram is merely a simplified representation of the problem that communicates to yourself and others the forces you believe are relevant to the problem. The number and direction of the forces does NOT change regardless of the behavior of the elevator. Whether the elevator is standing still or going down while slowing down the two forces shown are always there. The relative 'sizes' of the forces does depend on the behavior of the elevator but the freebody diagram DOES NOT!

Activity: Develop a freebody diagram for the pygmy giraffe.

HW: Newton's Law's: FBD

Consider a skydiver falling from the plane before and after her parachute opens. Create a freebody diagram for each circumstance labeling your forces and being clear about any forces you choose to neglect and why.

HW: Newton's Law's: FBD

The Sky Crane was the model for delivering the Curiosity Rover to Mars. Here is a link to a National Geographic video describing this deployment process. Create a freebody diagram for the landing system at three points in the process. While the lander is falling with the parachute deployed, after it separates from the parachute ut before it deploys the Sky Crane, and finall as the Sky Crane is deployed. Your freebody diagrams should make it clear which forces are relevant at each point in the process and any forces that are neglected.

Newton's Laws:

We now have some sense of what a force is and how to communicate the forces we believe are acting on a particular object. Now we need to work our way towards an understanding of Newton's Laws. We will begin by considering what we expect to happen to an object in the absense of any forces and slowly build our way towards situations where there are multiple forces that are not balanced.

No Forces:

The philosophers among you may wonder whether it is possible to find an object in the universe which experiences no forces. Gravity is everywhere is this idea of an object experiencing no forces relevant. Galileo was one of the first who approached this sort of question through a process called iteration. In this context we will consider an object which has only one force acting on it and then consider what we observe to happen as that force gets smaller and smaller approaching nothing. Image a tennis ball on earth. If you let it go it drops rapidly which we associate with the force due to gravity. If we take the same tennis ball out into the space somewhere along the moon's orbit and release it what do you expect? Now imagine carrying the same tennis ball out past Pluto into deep space far from any object and releasing it. What is the logical extension of this pattern?

Stationary Object:

I trust that you are confident that a stationary tennis ball that experiences no forces will remain stationary. Even though you have never seen such a thing this makes sense.

Moving Object:

What if we now consider the same tennis ball in deep space but I give it a little push and step back? While I am pushing it there is a definite force but once I stop 'touching' it my freebody diagram is empty again. Are you confident that the tennis ball will keep moving steadily without changing direction in the absence of forces?

First Concept:

This leads us, as it did Galileo and other early scientists both before and after, to the idea that, in the absence of forces on an object, it will continue to do what it was doing. Whether that is being stationary or moving steadily it will keep doing what it was doing when all the forces are removed.

Stationary Object with Forces:

Does this mean that a stationary object is feeling no forces? Of course not. A book sitting on a table is stationary and yet there are clearly forces (at least 2) acting on it. The freebody diagram for a book looks very much like the bat above except the upward force is a force from the table. What is it that is similar about a book with these two forces and the same book far away with no forces acting on it?

Balanced Forces are Equivalent to No Forces:

We reconcile this situation by suggesting that if the two forces on the book are 'balanced' (the up force is the same size as the down force) then that is equivalent to there being no forces on the book. This is only true in the sense the the behavior of the book as a whole is equivalent. The internal experience of the book is different since in one case it is being 'squeezed' and in the other case it is not. Those are internal experiences.

Moving Objects with Balanced Forces:

What do you think will happen if the object is already moving while subject to balanced forces? Why do you think this? All things in our world seem to come to a rest if you leave them alone. This led the Greeks and other scientists long ago to imagine that being at rest was the natural state of every object. Without a clear idea about forces and their effect on objects this is reasonable but incomplete.

Galileo used the iterative process described earlier to address the question of balanced forces on moving objects. take a puck which is just like a book when it's sitting still. Two balanced forces are acting on it in the vertical direction. Slide the puck along a table. It will come to a stop after some distance - why? Make the table smoother and repeat the experiment -- what changes? Cover the table with a very light oil and repeat. Imagine the table can be as long as needed and the friction can be reduced to nothing. What is the logical extension of this experiment? The moving puck, with balanced forces, continues to move forever.

Net Force: F_net:

We use the term 'net force' or F_net to describe the collective effect of all forces acting on an object. It is important to understand that F_net is a description of the collective effect of forces and NOT a separate force on it's own. F_netshould not show up as a force on your freebody diagram.

Newton's 1st Law:

All of these questions and explorations are intended to get you to think about whether you believe your experiences justify the following observations about the behavior of objects with no forces or no net forces acting on them. Notice the frustrating differences between these different versions of the 1st Law taken from different physics texts. You may well ask why we don't just quote Newton? We don't because the specific language he used, even in the first English edition, has different meanings for the words than we currently have.

First law

If no net force acts on a particle, then it is possible to select a set of reference frames, called inertial reference frames, observed from which the particle moves without any change in velocity. This law is often simplified into the sentence "An object will stay at rest or continue at a constant velocity unless acted upon by an external unbalanced force".

Newton's First Law states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force.

First law: In an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.

Which version of Newton's First Law will you remember?

Newton's First: Important Features

i: If F_net = 0 (include when there are no forces) then the object remains stationary or in motion in a straight line without change.

ii: If the object remains stationary or moves in a straight line without change then F_net = 0.

iii: If F_net = 0, from your freebody diagram, and the object changes it's motion then Newton's Laws do NOT apply from the perspective of this observer.

HW: Newton's Law's: FBD

A block is hung in the middle of the elevator by a string from the ceiling. Create a freebody diagram showing the forces acting on the block. In which of the following circumstances does Newton's 1st Law tell us that F_net = 0? It's perhaps slightly premature but in the other cases which force(s) do you think are the largest?

i) The elevator is at rest.

ii) The elevator is moving upward at increasing speed.

iii) The elevator is moving upward at decreasing speed.

iv) The elevator is moving upward at constant speed.

v) The elevator is moving downward at decreasing speed.

vi) The elevator is moving downward at constant speed.

[Fs=Fg, >,<,=,>,=]

Assignment: HW: Newton's Law's: FBD

Turn in the various (3) homework problems in this breadcrumb. For these problems please use the problem solving format described in the Concepts breadcrumb.

Assignment: Reading

Go on to Newton's Laws 1D II breadcrumb to move to calculational applications of Newton's Law's in 1D.