Lesson 4 – Introduction to Classical Mechanics - Building Momentum

Now that we have covered motion, it's time to talk about Momentum. However, before we move on let's review what we have learned thus far:

• Matter: Anything that has Mass and Volume
  
• Volume: The amount of three-dimensional space a substance (solid, liquid, gas, or plasma) or shape occupies or contains. Volume is used to determine the Density of an object.
  
• Mass: Mass is the amount of material or matter an object is made up of. Ability to resist motion when acted on by a Force due to the quantity of material in the object. Represented by (m) and the unit of measurement is the Gram (g)
  
• Density: Tells us how tightly crammed together matter is. density = mass / volume
  
• Force: Any type of influence that causes a free body to undergo an acceleration
  
• Displacement: The shortest path between two points and thus is always less than or equal to distance. Work = Force * Displacement
  
• Speed: The rate (distance & time) at which an object moves relative to some defined reference point of view. speed = distance / time
  
• Velocity: Velocity = Displacement / Time
  
• Acceleration: The rate at which an object changes its velocity. Acceleration = ∆ Velocity / Time
  
• Newton's Three Laws of Motion:
  
  • First Law
    
    • Unless acted on by an outside force, an object is at rest and stays at rest.
      
    • Unless acted on by an outside force, an object moving with uniform velocity continues to move at that velocity
      
  • Second Law
    
    • If an object of mass m (kilograms) is acted on by a force of magnitude F (newtons), then the magnitude of acceleration a (meters per second squared) can be found by:
      
      • a = F / m
        
      • F = ma
        
  • Third Law
    
    • Every action is attended by an equal an opposite reaction.
      
    • If Object A exerts a Force F on Object B then Object B exerts a Force F on Object A.
      

Momentum

Classical Mechanics describes the behavior of objects in motion. And any object moving that has mass, has momentum and energy. As objects collide, their energy and momentum change. Momentum is the product of an object's mass and velocity. Since the standard unit of mass is the gram (g), and the standard unit of speed is the meter per second (m/s), Momentum magnitude is expressed in gram-meters per second (g * m/s).

Example:

• Mass of object in motion increases by 10, Speed stays constant, therefore Momentum increases by a factor of 10
  
• Speed of object in motion increases by 10, Mass stays constant, therefore Momentum increases by a factor of 10
  

Momentum is a vector, not only does it have magnitude it also has direction. The direction is based on the direction of the velocity of the mass.

Impulse

Welcome back to the Millennium Falcon. Han Solo just started yelling about how the Empire has secretly disengaged the Hyperspace engine and you need to get to Cloud City to see Han's buddy Lando! Han starts blabbering about the Impulse engines. You sit there and look at Han and say what is Impulse??

Impulse produces a change in velocity and is equivalent to the change in momentum…let's see why.

Momentum of an object can change due to the following three reasons:

1. Change in the Mass of the object
   
2. Change in the Speed of the object
   
3. Change in the Direction of the object
   

When we consider the second and third reasons together we are really considering a change in velocity.

Impulse = Force * Time

Force = Mass * Acceleration

Therefore:

    Impulse = (Mass * Acceleration) * Time

Acceleration = (∆ Velocity / Time)

Therefore:

    Impulse = (Mass * (∆ Velocity / Time) ) * Time = Mass (∆ Velocity) = Change in the Momentum

Collisions

When two object's paths cross at the exact same time and they are in relative motion which causes them to strike each other, it is called a collision. According to the Law of Conservation of Momentum, the total momentum contained in two objects is the same after a collision as before. In an ideal system the characteristics of the collision do not matter since there is no friction or imperfections. In addition, in an ideal system, the total system momentum never changes unless a new mass or force is introduced.

The Law of Conservation of Momentum only holds true in a closed system, a system in which the total mass remains constant, and no forces are introduced from the outside.

Lesson 3 - Introduction to Classical Mechanics - Moving On Up

Now that you know about the Force lets quickly review the formulas and definitions we have learned before moving onto Newton's Laws of Motion!

• Matter: Anything that has Mass and Volume
  
• Volume: The amount of three-dimensional space a substance (solid, liquid, gas, or plasma) or shape occupies or contains. Volume is used to determine the Density of an object.
  
• Mass: Mass is the amount of material or matter an object is made up of. Ability to resist motion when acted on by a Force due to the quantity of material in the object. Represented by (m) and the unit of measurement is the Gram (g)
  
• Density: Tells us how tightly crammed together matter is. density = mass / volume
  
• Force: Any type of influence that causes a free body to undergo an acceleration
  
• Displacement: The shortest path between two points and thus is always less than or equal to distance. Work = Force * Displacement
  
• Speed: The rate (distance & time) at which an object moves relative to some defined reference point of view. speed = distance / time
  
• Velocity: Velocity = Displacement / Time
  
• Acceleration: The rate at which an object changes its velocity. Acceleration = ∆ Velocity / Time
  

 

Newton's Laws of Motion

 

In Physics, like all other subjects there have been some truly great men and women. Sir Isaac Newton is one member of this group in the world of Physics.

Newton came up with three laws which apply to the motion of objects in classical physics. It is important to note that these laws do not apply when speeds approach the speed of light or where extreme gravitational fields exist (like in a black hole).

First Law

• Unless acted on by an outside force, an object is at rest and stays at rest.
  
• Unless acted on by an outside force, an object moving with uniform velocity continues to move at that velocity
  

Second Law

If an object of mass m (kilograms) is acted on by a force of magnitude F (newtons), then the magnitude of acceleration a (meters per second squared) can be found by:

• a = F / m
  
• F = ma
  

Third Law

Every action is attended by an equal an opposite reaction.

If Object A exerts a Force F on Object B then Object B exerts a Force F on Object A.

Lesson 2 - Introduction to Classical Mechanics - Use the Force!

Last lesson, What's the Matter, we introduced the concept of matter and some of its most important characteristics. Some of the definitions and formulas from the last lesson are available below.

• Matter: Anything that has Mass and Volume
  
• Volume: The amount of three-dimensional space a substance (solid, liquid, gas, or plasma) or shape occupies or contains. Volume is used to determine the Density of an object.
  
• Mass: Mass is the amount of material or matter an object is made up of. Ability to resist motion when acted on by a Force due to the quantity of material in the object. Represented by (m) and the unit of measurement is the Gram (g)
  
• Density: Tells us how tightly crammed together matter is. density = mass / volume
  

 

Well if you are reading this then I assume you are ready to learn about the Force. In order to better describe Force and to pay homage to Star Wars, I will offer an example from a relative setting.

You are hanging out in the Millennium Falcon and whatever they use to make artificial gravity has been damaged so that things are floating around in the cabin. In front of you are two floating objects, a blaster helmet and a small droid restraining bolt. You decide in your boredom while the Wookie is repairing the ship to see what would happen if you flick both the bolt and the helmet with your finger in the exact same manner. The bolt flies across the cabin and hits the wall very quickly. However, the blaster helmet takes several minutes to float across to the cabin before it hits the wall. Your finger imparts a force to the bolt and the helmet for only a moment, but the force has a different effect on each object.

The effect of an applied Force on a given mass can be measured. In addition, a Force can be measured by the amount of deflection or distortion it produces in an object that is elastic, such as a spring. Force is typically measured either by a Meter or in comparison to another Force.

Example:

The bathroom scale is a good example of an everyday Force meter. It measures the amount of Force exerted on it in terms of (units of) weight.

Scalars and Vectors

The Mass of an object is a scalar and the unit of measurement is the Gram (g). Scalars are single units of data. In the case of Mass, it has a magnitude which can be measured but has no other quality. Hence, it is a scalar. Force on the other hand is a vector and the unit of measurement is the Newton (N). This is because Force has more than one quality to be measured, magnitude and direction.

If we were to chart Force it would look like a line segment with an arrow, where the line segment length would indicate magnitude, and the arrow would indicate direction.

Question: So what exactly is Force?

Answer: Force is any type of influence that causes a free body to undergo an acceleration.

Displacement

Now that we know about Force, its time to be able to define its type of effects. However, before we can do this we need to learn about the concept of displacement. Displacement is also known as distance. However, there is a slight difference between the two. Displacement is actually the shortest path between two points and thus is always less than or equal to distance. Displacement is a vector quantity because it has magnitude (usually expressed in kilometers) and direction.

Displacement is important because physicists measure work by multiplying the force times the distance over which the force is applied. work = force * displacement

We will return to this formula and concept of work later on.

Speed

One effect a force can have is to change the speed of an object. Speed is the rate (distance & time) at which an object moves relative to some defined reference point of view. The reference frame should be considered stationary (this is relative however). speed = distance / time

Speed is a scalar as it only has magnitude and no other qualities such as direction. The standard unit of speed is meters per second (m/s).

Question: What do you mean the reference point is stationary?

Answer: Lets pretend you are standing on the Earth and watching the movement of the Moon. You measure its speed at a relative stationary point. The point is relative because you are only stationary relative to the Earth, but the Earth is in fact moving as well.

Velocity

Velocity is made up of both magnitude and direction and hence is a vector quantity. The direction can be defined in one or more dimensions. For our purposes currently we will limit the dimensions to three. The formula for Velocity is: Velocity = displacement / time

Acceleration

Acceleration is the rate at which an object changes its velocity. This can occur due to a change in speed, a change in direction, or a change in both. If velocity does not change, then there is no acceleration!

Like velocity it can take place in multiple dimensions and can, but does not have to follow the same direction as the velocity vector. Acceleration magnitude is expressed in meters per second per second. (m/s^2). Acceleration = ∆ Velocity / time

Question: What do you mean per second per second or seconds squared?

Answer: Lets say we have a car that can go from 0 to 60 miles per hour in 5 seconds. A speed of 60 miles per hour is roughly equivalent to 26.8 m/s. Then to calculate the acceleration magnitude: a = 26.8 m/s / 5 s = 5.36 m/s^2

Lesson 1 - Introduction to Classical Mechanics - What's the Matter?

Everyone at some point in time has heard stories about when people thought the world was flat, the Earth revolved around the Sun, or how the Earth's orbit made music. While those stories may seem silly today, they were ironclad truths in their time. One thing they all had in common however, was that since ancient times, people have been trying to explain the world around them. Ironically enough with each new period of enlightenment, like the television show Lost, more questions followed.

The world known to the earliest physicists was much smaller and less complex than the world today (One may joke that its all relative. If you don't get this joke yet do not worry, you will eventually). This is because so much less was known about it. Think of when you were a very small child. The world around you seemed so simple in comparison to the world as you know it now.

Early physicists questioned the world around them and attempted to develop mathematical models (theories) to explain and predict how objects in the world interacted. These objects were eventually said to consist of Matter.

Matter has had variations in definition over the years as well as when applied to various fields of study. However, for our current purposes:

• Matter: Anything that has Mass and Volume
  
• Volume: The amount of three-dimensional space a substance (solid, liquid, gas, or plasma) or shape occupies or contains
  
• Mass: Ability to resist motion when acted on by a Force due to the quantity of material in the object
  

Question: So what has Matter?

Answer: Everything around us!

Examples:

1. You are made of matter
   
2. Your car is made of matter
   
3. The Earth is made of matter
   
4. Pumpkin pie is made of matter
   

Question: What is the relationship of Volume to Matter?

Answer: Volume is a measurement of the space occupied by an object. Volume is used to determine the Density of an object. Density tells us how tightly crammed together matter is. Density has many properties that assist us in our analysis of the world.

Examples:

1. Wood is generally used for boats because it floats in water. This is because wood is less dense than water.
   
2. If we want a balloon to rise it is important to fill it with a less dense gas than air such as helium
   
3. Given the volume of an object, the density will give us the mass. This is because the formula for density = mass / volume
   

Question: How does mass resist motion when acted on by a force? What is mass?

Answer: Mass is the amount of material or matter an object is made up of. Mass is the measure of heftiness of the material of an object. Therefore, think of kicking a balloon. The balloon does not resist moving due to the implied kick of your boot. This is because the balloon does not contain much mass. However, if you kick a rock the same size as the balloon it will not move nearly as much. This is because the rock has a much greater mass.

Mass has no direction. To measure mass on earth we need special tools. An easy method to calculate mass without special instruments would be to calculate weight. Weight is the gravitational pull on an amount of mass.

If we weigh two items on earth (under the same gravitational pull), the more massive object will always weigh more.

Another way to measure mass would be to use a triple-balance beam scale

The usual symbol for Mass is m and its unit of measurement is the Gram (g).

In the next post we will make like the Jedi's and use the Force!

An Introduction

We all remember our first lessons in science and mathematics. Some with fonder memories than others. However, for most of us it has been a mishmash of instructors who either do not understand the subject matter or those who do not understand how to teach. Rarely are we fortunate enough to find an instructor in science or mathematics that is capable in both areas.

When I was young I was fortunate enough to have excellent instructors. However, this all changed when I attended college. In college I found that the majority of instructors either did not know how to teach or sent their assistants to regurgitate their latest book. In the end I realized that I would be on my own to learn the subject matter.

Learning science and mathematics on one's own is not an easy task. Both subject areas are vast as they have evolved in parallel with human society branching in various directions along the way. I am in no ways an expert or claim to have any expertise in science or mathematics. However, I am offering a unique learning perspective from this blog. In this social media experiment, we will learn what many consider as the most dreaded of subjects together, physics. We will start from the building blocks and build our way towards a greater understanding of the world around us and eventually the universe. I will try an keep each lesson concise and offer examples that everyone can understand.

You may ask who this blog is for.

• You turn on the science channel at night and watch the latest special hosted by Michio Kaku and now want to play around with some ideas in your spare time
  
• Your child just asked you what is gravity and you said it makes things fall down
  
• You just read Simon Singh's totally awesome book, Big Bang, the Origin of the Universe and realized there is so much more to know
  
• You are a student or a parent of a student struggling with the fundamentals of science and mathematics
  

Well I hope you join me in a fantastic journey and a wonderful social experiment. Who knows, a little understanding of the world around you just might change your life!