Notes
These are the topic areas covered in the notes for this unit.
Basics of Forces (Qs and Ps for this section) (back to the top)
So far you have studied the aspects of motion of an abject; displacement, velocity, and acceleration. What causes an object to slow down, to turn a corner, to speed up? The cause of any change in motion is called a force.
Force - an action which tends to cause a change in motion. A push or pull.Forces come in two basic flavors; contact forces and field forces.
Contact forces occur as the name implies - when contact between two object is present. You cannot push your book across the table without contacting the book. A tennis racket cannot change the motion of the ball unless it contacts the ball.
By contrast, a field force tends to change motion at a distance. The earth exerts a gravitational force on an object even when the object is not touching the earth. Two magnets will affect each other before they come into contact.
Until a few years ago scientists believed that all forces could be categorized into five classes:
This means that all forces, whether they are contact forces or field forces, are either a form of electrical force or gravitational force. When you push your book across the table it is the electrons in the atoms of your skin which are repelling the electrons in the atoms of the book. If you think contact is "actually touching" then you never really touch the book. The effect is the same no matter how you define it.
Force tends to cause a change in motion of an object. The definition of a force is an operational definition; it is defined by what it does. Remember that changes in motion are measured as acceleration. Forces tend to cause acceleration.
The unit of measurement of force is the Pound (lb) in the English system and the Newton (N) in the metric system. You are most familiar with pounds, like the measurement of your weight. Newtons will be new for you.
To get an idea of the size of a Newton, one Pound is the same force as 4.45 Newtons. Roughly speaking a Newton is a quarter Pound. A 100 lb person would weigh 445 N. That's one reason we don't use the metric system to express weight. No one wants to weigh 445 of anything.
Notice that the phrase "tends to cause" has been used because, as you will see later, a single force may not cause a change. It is only when the total of all the forces acting on an object has a non-zero value that the object will change its motion.
Now here's an interesting fact. Forces never operate alone. They always occur in pairs. Isaac Newton stated this as one of his basic laws back in the 1600's. We often refer to these two forces as action/reaction forces. You can't tell which is the action and which the reaction, but it doesn't make any difference.
Simply stated I can not push on you without you pushing back on me. Our pushes are the same size (magnitude) , bit they are in opposite directions. I push on you. You push on me.
If you throw a ball against the wall the ball pushes on the wall, and at the same time, the wall pushes on the ball. The wall pushes on the ball with the same amount of force that the ball pushes on the wall. One of the forces is on the ball, the other force is on the wall.
This is one of the most difficult concepts for students to accept.
Think of this example: A small car has a head on collision with a large truck. Does the car get "hit harder" than the truck?
Your first inclination would be to say "yes, the truck hits the car harder than the car hits the truck". Guess again. There is no doubt that the car will have more damage than the truck, that the people in the car may suffer more injuries than for those people in the truck, but it is not because the car gets "hit harder". As we will see in the next unit, the reason for the inequity in the collision is due to the smaller mass of the car. They both hit each other with the same force.
When a golf club hits a ball, the force of the club on the ball is the same amount of force the ball exerts on the club.
When a big senior runs into a small freshman in the hall the force of the senior on the freshman is the same as the force of the freshman on the senior.
When you stub your toe on a table leg the force of your toe on the table is the same force as the table on your toe.
Think about it, let it sink in. Whether you believe it or not, it's true.
Force Diagrams (Qs and Ps for this section) (back to the top)
A force diagram is drawn to show the relative magnitudes and directions of the forces involved in a situation. The actual object the forces are applied to is not relevant in the diagram.
For simple situations the force diagram may seem to be overkill, but for more complicated situations it is an invaluable tool.
Many students get into trouble when solving force problems because they did not take the time to draw a diagram. They usually are confused about the forces which are involved and never get a good picture of the situation.
As with solving motion problems, the first step of identifying the variables is a very important one. Translating from words to pictures, then numbers is not accomplished by many without practice. Just watching others solve the problem does not create the skill of being able to solve the problem for yourself.
Let's look at a simple example: You push horizontally on a couch while your friend pulls on it in the same direction. Let's also say that you apply twice the force as your friend.
The force diagram would look like this:
Notice how
In another case: You and an opponent each kick a soccer ball at the same time from opposite directions. Your opponent's kick is larger
Did you picture a force diagram like this?
Again
What if the forces are not along the same line?
An airplane heads due north in a region where the wind is blowing in an easterly direction.
Here the diagram shows the vectors relative to the coordinate system given.
Notice,
Be conscious of forces implied by the wording of the problem. Is the weight of the object relevant? Is a force applied by a rope, string, or wire? (tension), is there friction in the problem?
Draw a force diagram for a picture which is hung by means of a wire stretched across the back of the picture and then hung on a nail in the wall. The angle between the two sections of the wire is 150°.
Look at the picture and decide what point the forces have in common; looks like the nail in the wall. What are the forces on that point (nail)?
Here's the force diagram:
Notice
Here's a final situation: You push your calculator across a table with a force directed 20° below the table.
Before you draw a diagram, list all of the forces involved, there are four.
Here's the diagram:
As before
Drawing a force diagram is an important step in solving problems.
Don't be in a hurry and omit it.
Addition of Forces (Qs and Ps for this section) (back to the top)
Single forces each tend to cause a change in the object's motion, but it is the total force which determines what happens to the object. Finding the total force (or the net force) can be simply stated, but may be time consuming to do.
Total Force (also called the net force) = the sum of the individual forces
This definition does not seem too involved until you remember that forces are vector quantities. Adding vectors is not the same as adding scalars. You have to account for the direction of the forces involved. The following situations will show the general procedures followed when adding force vectors. If you think about the relative directions of the forces you are adding the process should become apparent. No individual step in any of the processes is difficult, but in some cases the procedure requires a combination of many steps.
For forces in the same direction
If two (or more) forces are in the same direction you can add the magnitudes of the forces together. The direction of the total force would be the same direction as the original forces.
EX. You push horizontally on a couch with a force of 200 N while your friend pulls on it
in the same direction with a force of100 N.
In this case the force diagram is a little overkill, but here it is with the solution.
For forces in opposite directions
If two (or more) forces are in opposite directions you first need to assign a pos/neg value to each force. Which is which does not matter as long as forces in the same direction have the same sign and forces in opposite directions have opposite signs.
Once the pos./neg. have been labeled you can add the magnitudes of the forces together. The direction of the total force would be determined by the sign of the result.
EX. You and an opponent each kick a soccer ball at the same time from opposite directions. Your force is 75 N and your opponent's kick is 125 N.
Again, the force diagram is not strictly required, but here it is with the solution.
For forces in perpendicular directions
If two forces are in opposite directions you first need to draw the force diagram to get the information correct. To add the vectors you must:
Be sure to express the answer in similar terms as the original vectors.
This is a multistep process, don't shortcut it.
EX. An airplane heads due north in a region where the wind is blowing in an easterly direction.
If the plane's engines push it forward with a force of 2.5 E3 N while the wind's force is
1.0 E3 N, what is the total force on the plane?
Here are the steps of the solution:
For forces in nonperpendicular directions
This process is similar to adding forces which are perpendicular, but you must resolve each vector into its components first.
The process:
First, Draw the force diagram.
Second, resolve the boat's thrust into components.
Third, find the total X and Y components.
Fourth, use Pythagorean formula and Tangent function to find the magnitude and angle for the total force.
Last, express the answer in similar terms as the given information.
The force on the boat is 1512 N, 6° upstream from straight across the river.
For addition of more than two forces
In cases where there are more than two forces the procedure to follow is the same as that for nonperpendicular forces. Just remember to resolve any force which is not on an axis into its components first.
In Summary:
|
|
|
| In the same direction | Just add them together |
| In opposite directions | Assign +/-, add them together |
| Two at right angles | Use Pythagorean formula for magnitude, and tangent for direction |
| Two at non-right angles | Resolve forces into components, find totals on each component axis, use Pythagorean formula for the final magnitude, use tangent for the direction |
| More than 2 forces | Resolve forces into components, find totals on each component axis, use Pythagorean formula for the final magnitude, use tangent for the direction |
Equilibrium (Qs and Ps for this section) (back to the top)
If the forces applied to an object create a total (net) force of zero then the object will not change it's motion - it is in equilibrium. If the object is stationary it is called static equilibrium. If the object is in motion it is called kinetic equilibrium.
Equilibrium - a condition of unchanging motion created by a zero total force.
If an object is in equilibrium it is a clue to the nature of the forces acting on it. A single applied force cannot cause equilibrium unless it has a zero value, but then it isn't there in the first place.
It takes at least two forces to create a condition of equilibrium. If there are only two forces then they must have the same magnitude and be in opposite directions. This is the only way in which their sum could be zero.
Action and reaction forces cannot cause equilibrium. Since they are operating on different objects, they cannot be added together. Some third force may be present to oppose an action or reaction force.
If you kick your locker, your locker exerts an equal force back on your foot. Your foot changes its motion; it stops. The locker doesn't move because your kick did not overcome the other forces holding the locker in place. If it does, the front of the locker would start to move (a dent).
Knowing an object is in equilibrium allows you to find unknown forces. For example, if it is known that the forward force on your car is 2000 N, but the car is moving down the road at a constant velocity (equilibrium) then the frictional force on the car must be 2000 N in the opposite direction.
Suppose two elephants are pulling on a shirt, one exerts a force of 35 N [0°] and the other a force
of 55 N [90°]. What force would you have to apply to the shirt to keep it atationary?
Keeping it stationary means the shirt would be in equilibrium. The total of the three forces (the two elephants and you) would have to be zero.
First, find the sum of the forces from the elephants.
Your force would have to be equal in magnitude to the elephant's combined force and opposite in direction.
The object of many aspects of engineering is to create equilibrium. Knowing the basis of the cause of equilibrium and how to calculate numbers in its application goes a long way to working in this field.
Gravity (Qs and Ps for this section) (back to the top)
The force of gravity is an attractive force between any two objects with mass. This was the view of Isaac Newton in the late 1600's. Albert Einstein modified the concept in the early 1900's, but the calculations based on Newton's work is still valid.
The force due gravity is a mutual force shared by the objects involved. It simplifies matters if we consider only two objects. Mathematically stated, Newton's universal law of gravitation is:
where: M1 and M2 are the masses of the two objects
R is the distance between the centers of the two objects, and
G is a constant = 6.67 E-11 N m2/Kg2
EX. What is the gravitational attraction between two students, each 70 Kg separated by 2 m?
By substitution: Fg = 6.67 E-11 N m2/Kg2 (70 Kg) (70 Kg) / (2 m)2
= 8.2 E-8 N This is a very small force.
It should become apparent that there is a difference between the force of gravity between two objects and the mass of the objects. In common terms the force due to gravity due to the attraction with the earth is called the weight of the object.
Mass is a characteristic of an object; it belongs to the object.
The only way to change the mass of the object is to change the object itself.
Weight is a force on the object. It happens to the object.
To change the weight of an object, change the environment the object is in.
Mass is measured in grams or kilograms (Kg). Weight is measured in Newtons (N).
When calculating the weight of an object on the earth's
surface you could use the formula above, but every calculation would share some common values.
The gravitational constant (G) would be the same for every calculation as well as the mass of the earth (6 E24 Kg) and the distance between the objects, the radius of the earth (6.4 E6 m). If you rewrite the equation with these common values
The equation simplifies to the following:
Fg = 9.8 N/Kg (m)
Notice the 9.8 N/Kg is a conversion factor, of sorts. You've seen this number before, remember? To calculate the weight of an object on the surface of the earth, just multiply the mass (in Kg) by 9.8.
EX. What is the weight, in Newtons of an 80 Kg person?
Fg = 9.8 N/Kg (80 Kg) = 784 N
As you have seen before, you can equate force measurements in pounds (the english system) with Newtons (the metric system).
4.45 N = 1.0 lb
EX. What is the weight, in newton's of a 140 lb person?
Fg = 140 lb ( 4.45 N / 1.0 lb) = 623 N
If you put these last two expressions together, you can relate the weight of an object in Newtons and its mass in Kg.
1.0 Kg weighs 2.24 lb
Notice that there is no equal sign in this expression because Kg (mass) and lb (force) do not represent the same type of quantity. Equations are only valid if both sides are the same type of quantity.
The universal law of gravitation can be applied to any two masses. It is also useful in determining how the weight of an object would change when placed on another planet or moon.
EX. How would be the weight of an object be different if it were on a planet half the mass of the earth, but with a radius twice as large?
Solution:
Friction (Qs and Ps for this section) (back to the top)
Friction is often used as a catch-all category for any thing which acts against the motion of an object. In reality it is a fairly narrow collection of reaction forces. In this unit we will restrict our discussion of friction to that which exsits between surfaces sliding against each other.
Sliding friction is a reaction force generated by the surfaces in contact. The microscopic view of the surfaces shows ridges and valeys which cause a bumpy ride as the surfaces slide across each other.
Friction always opposes motion, or the tendency of an object to move if it is at rest. The force of friction is parallel to the plane of the surfaces and depends on the following:
A fact surprising to most students is that the amouont of frictional force does not depend on the amount of surface in contact. Larger surface areas do not create more friction than smaller areas. The reason for this comes from the reasoning that any section of the larger surface is pressed into the surface less than a similarly sized region of the smaller area. As a result the product of the amount they are pressed together and the surface area are the same.
There are two types of friction:
At the right is a graph of the force applied to a stationary object until it moves at a constant rate. You can see a peak force is reached at the point where the object begins to slide. At this peak the applied force would be equal to the maximum force of static friction.
After the object is moving a smaller force is needed to keep it moving at a constant rate. Since the object is in equilibrium the applied force would be equal to the kinetic frictional force.
One of the factors determining the frictional force is the Normal force.
The Normal force (N) is the force applied by a surface on an object.The Normal force is always directed away from the surface and is perpendicular to the surface. The Normal force is also a reaction force. The harder the object pushes on the surface the harder the surface pushes back.
EX. What is the normal force on a 2.4 Kg book sitting on a table?
The roughness of the surfaces in contact is indicated by a coefficient of friction (m). The value of the coefficient is determined by the ratio of the frictional force to the Normal force. A higher value for the coefficient indicates a larger frictional force exists between the surfaces.
Since there are two distinct frictional forces, there are two different coefficients for the surfaces. As you might suspect the kinetic coefficient is smaller than its corresponding static coefficient.
The coefficients depend only on the roughness of the surfaces, not on the Normal force. If the normal force is increased, the frictional force is increased, but the coefficient remains the same. The only way to change the coefficient is to change the surfaces in contact. A limited table of coefficients is given below.
EX 1. A 30 Kg wooden crate sits on a cement floor where its ms = 0.7 and a mk = 0.4.
a) What is the maximum force of static friction?Solution:
b) What is the kinetic force of friction?
c) If a 70 Kg person sits on the crate,
1) would the coefficients of friction change, and
2) what minimum force, parallel to the floor, would it take to make the box move?
a) Fs = ms (N) where N = Fg = 9.8 N/Kg (30 Kg) = 294 N
= 0.7 (294 N)
= 206 N
b) Fk = mk (N) where N = Fg = 9.8 N/Kg (30 Kg) = 294 N
= 0.4 (294 N)
= 118 N
c) 1) No. The coefficient depends on the types of surfaces and they haven't changed.
2) The minimum force to move the crate would be equal to the static frictional force.
Fs = ms (N) where N = Fg = 9.8 N/Kg (100 Kg) = 980 N
= 0.7 (980 N) (the weight of both the person and the crate together)
= 686 N
The minimum force to move the crate would be 686 N
EX 2. A 1.7 Kg book lying on a table has a ms = 0.6 and a mk = 0.2. If it is pushed by a horizontal force of 9.0 N, will the book move?
What you need to know is, "Is the 9.0 N push more than the maximum force of static friction?"
Fs = ms (N) where N = Fg = 9.8 N/Kg (1.7 Kg) = 16.7 NThe maximum force of static friction is more than the 9.0 N push so the book will not move.
= 0.6 (16.7 N)
= 10.0 N