Forces
Concepts of Force and Mass
The common usage of the word force is a push or a pull. Consider the examples which illustrate the forces applied in each of these cases. In basketball, a player pushes a ball in the direction he wants it to go. The tow bar pulls the water skiier in the direction of the boat. A rock climber is pulled down by gravity, however, she controls her fall using the upward tension in the rope. Most of these are examples of contact forces, since they come as a result of the physical contact between two objects. There is one example of a noncontact force or actionatadistance force; and example of this is the force of gravity. The earth pulls you to the ground even if you are not in contact with it. Since forces are vectors, they have both magnitude and direction. In each of these examples, the arrows indicate the direction of the applied force. The length of the arrow identifies the magnitude.
Many people use the words mass and force interchangeably. However, a large object, for example, a boulder, contains a large amount of mass, however, it takes a great amount of effort to get it to move. Mass is actually about what it takes to get something to move. It is said that mass is the ability of an object to resist a change in its motion. Mass does not change. It does not matter if an object is on earth or on the moon; the object's mass remains the same. Additionally, mass is independent of volume. A kilogram (the measurement of mass) of feathers may take up more volume than a kilogram of lead, but each have the same mass. Additionally, where force is a vector quantity, mass is a scalar quantity. This means that mass has no concept of direction.
History of Forces
The study of forces and motion associated with them began with Aristotle.
Aristotle identified the concepts of natural motion and violent motion. Natural motion is the motion an object would have if left alone and unsupported. For example, a rock would fall. Natural motion also applies to things that move upward. For example, smoke will rise. Opposite to this is violent motion. This is the motion that occurs due to some outside influence. In the case of throwing a stone upward, you are applying an upward force causing a change in motion. Aristotle came up with the general statement:
For violent motion, the speed of the moving object was in direct proportion to the applied force.
This means that if you stop pushing an object the object will stop moving. He also stated that a heavier object will fall faster than a smaller object. Aristotles views on motion were considered the basis of all knowledge well into the 1700s.
Galileo as a part of his studies of planetary motion spent a great deal of time studying motion. In his studies, he determined that Aristotle was wrong when the said that heavier objects fall faster than lighter ones. Galileo did a number of studies related to Aristotle. He noted that sometimes it is important to slow motion down in order to better understand it. To do this, Galileo used a ramp in which he limited the friction, to study the motion of a ball as it rolls down. He theorized that an object would roll to exactly the same height on another ramp as it did on the first regardless of the angle of the first and second ramps.
What do you think would happen to the marble if it rolled down the ramp and then continued along a straight, level, frictionless surface?
It was Galileo that developed the concept of inertia. Inertia is the ability to resist a change in motion. In other words, the more inertia something has the more it is able to resist a change in its motion. Looking at our definitions thus far, mass does the same thing. This means that mass is a measure of an object's inertia. The greater the mass the greater the inertia. Galileo's study of intertia led him to his Prinicipal of Inertia that states:
No force is required to maintain motion with constant velocity in a straight line, and absolute motion does not cause any observable physical effects.
In the 1th century, Isaac Newton built on the work of Galileo and developed three important laws that deal with force and motion. As a whole they are called "Newton's laws of motion" and provide the fundamentals of our understanding of the effects that forces have on the motion of an object.
Newton's First Law of Motion
Newton's first law or the Law of Inertia is a general statement of the work of Galileo. Consider the game of ice hockey (figure below), an environment in which friction is neglible. (a) If you do not hit a stationary puck, the puck will remain at rest in its original location. (b) If you hit the stationary puck, it will move along the ice at a constant velocity once it leaves the hockey stick. In reality it will slow down slightly because of the small amount of friction between the puck and the ice. (c) The only change in the puck's motion occurs when the puck is in contact with the hockey stick. (d) Now if a second player hits the puck in a different direction, the puck will begin to move in a direction between the original direction and the direction of the second force.
This description of motion is fundamentally detailing Newton's First Law of Motion:

An object continues in a state of constant velocity unless acted upon by an outside net force. 

The term "net force" is the vector sum of all the forces. In other words, you have to take into account all the forces that act on an object in order to determine whether the net force will cause a change in motion, how much a change of motion will occur and in what direction the object will move. If the net force is zero, that means that the object will continue at a constant velocity, that is in a straight line at a constant speed. No net force is required for an object to maintain its velocity. If the net force is nonzero, the object will change its velocity (change its speed and/or direction). The most important point about net force is the cause of a change in velocity not sustaining it. Mathematically, a net force is identified by ΣF, where the Greek capital letter Σ (sigma) means vector sum.
Free Body Diagrams
Free Body Diagrams are pictures of objects or a representation of an object that have arrows that denote all the forces that are acting on it. Look at the example below:
In order for an airplane to fly, the lift force must be greater than the weight and the thrust must be greater than the drag. If the airplane is flying at a constant altitude, the lift must balance the weight. If it is increasing altitude the lift must be larger than the weight and the thrust must be larger drag. Decreasing altitude requires less lift but higher thrust. Landing requires higher drag than thrust. Remember only the forces that ACTUALLY act on the object should be included in the freebody diagram. For details review the examples on the left of this section.
Newton's Second Law of Motion
Newton's first law says that if no net force acts on an object, its velocity will not change. The second law deals with what happens when there is an unbalanced net force. If we go back to the ice hock example in section, the only time motion changes is when a stick is in contact with the puck. Remember that acceleration is equal to the rate of change of velocity. This means that if velocity changes, the object is accelerating. Based on this, it is the contact with the stick that creates the force and therefore the change in acceleration. Now consider a case where a second player hits the same puck with twice the force as the first person. The greater force will produce a greater acceleration. If you ignore friction and wind resistance, the acceleration of the puck is directly proportional to the force. So if you apply twice the force to the same object, the acceleration will be twice as much.
We discussed earlier that mass has to do with inertia which determines how much an object resists a change in motion. So what happens when you apply the same force to two objects of different masses. Consider the extremes, if you apply a force to an ice hockey puck, you will see an significant acceleration; however, if you apply that same force to a large SUV, you are going to get little if any acceleration. This means that acceleration is inversely proportional to the mass of the object. Newton's Second Law of Motion states

The acceleration of an object is directly proportional to and in the same direction as the applied net force and inversely proportional to the mass of the object.
a=ΣF_{net}/m or ΣF_{net}=ma 

The unit of force is kg•m/s^{2 }= newton (N). This makes complete sense since net force is equal to mass times acceleration and mass is in kilogram (kg) and acceleration is in meters per second squared (m/s^{2}). It is very important to remember that only the forces that are actively acting on the object are used in determining the net force and the corresponding acceleration.
1D Motion Example:
A 905.5 kg dragster is pushed to the starting line by 4 people. The first person applies a force of 400 N, the second a force of 300 N, the third a force of 250 N and the third applies a force of 350 N. Everyone is applying their forces in the same direction. There is a frictional force of 100 N between the dragster's tires and the road. Find the acceleration of the dragster and the velocity with which it reaches the starting line 40 m away if it starts from rest. (Click on the image to see and hear the problem solving process)
2D Motion Example:
Two boys are playing ice hockey and head for a stationary 0.25 kg puck reaching it at the same time. The first boy hits the puck with a force of 10 N at an angle of 20° above the horizontal. At the exact same time the second boy hits the buck with a force of 12 N at an angle of 35 degrees below the horizontal. What is the net force and acceleration of the puck? Include both magnitudes and directions.
Newton's Third Law of Motion
We have discussed the first two of Newton's laws. We know that a nonzero net force means that an object is going to change velocity (speed or direction), however, many times an object is moving at a constant velocity or is at rest. How did Sir Isaac Newton explain this phenomena? This is the basics behind his third law, which states:

If object A exerts a Force on object B, object B will exert a force equal in magnitude and opposite in direction.
F_{A}=F_{B} 

If an object is at rest, there cannot be a single force acting on it. If a single force is exerted on it, the object will move. This means there has to be at least one more force acting on it. Consider a box sitting on the floor, most people see that there is a clear force pulling the box towards the earth. They don't, however, see that there must be an opposing upward force; for without it, Newton's second law says that the box must accelerate downward. This opposing force is called the Normal Force or Support Force. These opposing forces are shown in the figure below:
Force pairs like weight and normal force always occurs between two objects. Consider the case of a basket of fruit on a table, there are three distinct force pairs: basket and table, basket and earth, table and earth.
These force pairs include:
 Basket pulls on the table and the table pulls on the basket.
 Earth pulls on the basket and the basket pulls on the earth.
 Earth pulls on the table and the table pulls on he earth.
There are, of course, pairs between each piece of fruit and between the fruit and the basket and the fruit and the floor.
Applications of Forces
There are many applications of Forces. This section describes many of those applications.
Weight and Apparent Weight
Weight is the gravitational force between a body and the earth. It is equal to
w = mg
where m is mass in kilograms and g is gravitational acceleration in m/s^{2}. It is important to remember that weight is a force and can be used as you would any other force. A scale does not read your weight; it actually tells the support force/normal force with which you pull the earth.
The normal force is not always equal to the weight. If an object has additional forces applied to it, the normal force will be equal to the sum of all the other forces. For example, if one box is sitting on a second, the normal force is equal to the sum of the weights of each of the boxes.
Apparent weight is the reading on the scale. Of course, when an object is at rest, the apprent weight is equal to the actual weight of an object. The apparent weight changes when an object is in motion. If a person is standing on an scale in an elevator, the reading on the scale will vary with the motion of the elevator. If the elevator is accelerating upward, the scale will read a higher value. If the acceleration is downard, the scale will read a lower than actual value. If the elevator is moving at a constant velocity, the scale will read the actual weight of the person.
Frictional Forces
In the Weight and Apprent Weight section, we dealt with forces perpendicular to the surface. This section will discuss forces parallel to the surface. When an object moves or attempts to move along a surface, there is a force parallel to the surface. This force is called a frictional force or friction.
Depending on the situation, friction can serve a good or bad purpose. For example, in a car, oil is added to ensure that the friction within the engine is reduced. On the other hand, friction between the tires and the road are imperative to ensure that a car moves. Tire treads are designed to provide as much friction as possible regardless of the condition of the road.
Friction was first studied systematically by Leonardo Da Vinci. He realized the how important friction is to machines. He also identified the difference between sliding and rolling friction.
Da Vinci stated two laws of friction 200 years before Newton defined the concept of forces. These laws are:
 the areas in contact have no effect on friction.
 if the load of an object is doubled its friction will also be doubled.


Leonardo observed how different objcts move with different amounts of difficulty. He surmised that this was a result of the roughness of the materials. Now, of course, Da Vinic realized that there must be two surfaces in contact. Friction is determined only by the make up of those two surfaces. What appears to be a smooth surface may not actually be so since it is the microsopic surface that determines the interlocking of the two surfaces.
Charles August Coulomb added to Da Vinci's second law of friction. He worked with this and came up with the AmontonsCoulomb law of friction.
Friction, surprisingly for some, depends only on the surface in contact and any additional supplied forces. It is not dependent on surface area or the mass of the object. The amount of friction between two surfaces is identified by the coefficient of friction, µ. There are two forms of friction, static and kinetic. Generally it takes more force to get an object moving than to keep it moving. This means that the coefficient of static friction, µ_{s}, will be larger than the coefficient of coefficient of kinetic friction, µ_{k}, for most surfaces.
Investigating an object undergoing friction (figure (a) to the right), an object is pulled with a horizontal increasing force. While the object is still at rest, the frictional force must be equal to the pulling force in order to maintain a net horizontal force of zero newtons. At some point, the pulling force is just going to exceed the static frictional force and the object will suddenly begin to move. After that sudden acceleration, the object will continue to slide. The object is now undergoing kinetic friction. A graph of the frictional force as a function of the force being pulled looks like the the figure to the right.
Coulomb friction, named after CharlesAugustin de Coulomb, is a model used to calculate the force of dry friction and is determined by the equation:
F_{f} = µF_{N}
where F_{f} is the frictional force, F_{N} is the normal force and µ is the coefficient of static or kinetic friction. Whether you use the coefficient of static friction, µ_{s}, or the coefficient of kinetic friction, µ_{k}, depends on whether the object is at rest or in motion, respectively.
Horizontal Example:
Inclined Plane Example:
Static Equilibrium
Static Equilibrium was originally studied by Archimedes. Much of his work was published in his work On the Equilibrium of Planes. Archimedes took the rather unsubstantiated and unscientific work of Aristotle and described the mathematics behind it.
Static equilibrium problems are problems dealing with objects that are at rest. This means that the net force in all directions is the same. Typically these types of problems involve hanging pictures and signs. To solve these problems, you set up a set of net force equations, first in the x direction and second in the y direction. This gives you a set of two equations each with two unknown variables. Solve these two equations like you would any other system of equations.
Multiple Object Force Problems
What happens when one object causes one or more other objects to move. Every object that is connected and travel with the same magnitude acceleration are considered a system. For the purposes of the class, any pulleys are frictionless and any strings or connections between objects are massless. This means that any string that passes over a pulley will have a constant tension over the entired length of the string.
For these types of problems, there are two approaches to solve the problems. The first is to develop free body diarams for each object in the system. This gives you a set of equations with a set of unknowns with them. The system of equations are then solved using substitution. The second approach is to treat the system as a single object and write a free body diagram including only the external forces on the system. Solve this equation and then solve for the internal tensions.
Example:
