Force

Table of Contents

• 1. Pushing an object, exerting a "force," makes it accelerate
  • 1.1. Objects do not change their motion without being pushed
  • 1.2. An object's mass gives it "inertia", a resistance to attempts to accelerate it
  • 1.3. Newton's second law tells us how force, mass, and acceleration are related
  • 1.4. If an object is not accelerating, the forces on it in each direction balance
  • 1.5. An object's weight is the force exerted on it by gravity
  • 1.6. An object resting on a surface is supported by a "normal" force balancing its weight
  • 1.7. If I try to slide an object along a surface, a friction force resists this
• 2. A force at an angle can be "resolved" into components using trig
  • 2.1. The trigonometric functions relate a triangle's side lengths to its angles
  • 2.2. Trig can be used to take a force at an angle and "resolve" it into components
  • 2.3. Force needs to balance in both components for an object to not accelerate
  • 2.4. Review of the force types we know so far
• 3. Exerting a force on any object makes it push back on you
  • 3.1. Force diagrams for multiple objects
  • 3.2. Newton's Third Law
  • 3.3. Reassembling a situation using the force diagram of each object
• 4. Review

1. Pushing an object, exerting a "force," makes it accelerate

1.1. Objects do not change their motion without being pushed

• What keeps a ball going through the air? Gravity only pulls down.
  
• What keeps the Earth going around the sun?
  
• A marble will roll for quite some time across a wood floor, but not as long on a carpetted floor. If I kept making the floor and marble smoother, would it eventually be able to keep rollinng forever?
  

1. If we see somthing at rest, or moving with constant velocity, we know that there is no force acting on it. Or, perhaps more accurately, we know that any force acting on it is balanced out by some other force; the car's engine provides enough force to balance the friction from air and the road.
   
2. If we see something that is accelerating, if we see any change in the motion of an object (speeding up, slowing down, or even changing direction), then we know that somethign is pushing that object.

1.2. An object's mass gives it "inertia", a resistance to attempts to accelerate it

• Construction in space is exceptionally dangerous because large objects the size of houses are moving around with no weight but a huge amount of inertia. Even a very slow-moving space station component can trap a person against another component and relentlessly crush him; the force he feebly exerts to resist this is insignificant compared to the mass of the thing he is trying to move.
  
• Astronauts are also very carefully tethered to the space station, because if they start floatinng away, no matter how slowly, there is nothing that can stop them; there is no friction in space.
  
• When Mr. Z. used to live in a house with 30 other MIT students, it was his job to bring home 12 gallons of milk every 5 days to supply the house. He would do this by loading them up in four shopping bags hung from a padded quarterstaff. When you have a hundred pounds of milk hanging from your shoulders, you don't feel much different when you're walking along the street (except for all the stares you're getting from passers-by), but you really do notice it when you try to stop and the milk want to keep going, or you start to turn and the milk keeps on turning.
  
• Mr. Z. once, as a child, tried to carry a pumpkin home from the store in a backpack, on a bike; the same thing happened to him, where he started to turn and the backpack kept on turning until he fell off the bike.
  
• Inertia is a very important concept in karate. If I want to stop someone who is running forward to attack me, it takes a tremendous amount of strength to sto them with a single punch or kick, but its not hard to get out of the way and let their inertia carry them past me. In fact, if I grab their arm and get their upper body moving even faster, their lower body will hang behind, due to inertia, and they will have to fall down. If they are clever, they will hit the ground in a roll, rather than a face-plant, so that there is less of a change in their motion and hence less force.
  
• I can move something quickly and trust that inertia will keep this from accelerating the things around it. For example, I can yank the tablecloth out from under a set table; there is little enough friction for little enough time that the inertia of each object keeps it roughly where it was. Or, i can pull out the bottom book from a stack, or knock out the bottom coin from a stack, and the rest of the stack will just fall down where it is.
  
• If you hit me with a sledgehammer, it would hurt a lot. However, if I were to lie down on a table and put a really massive object, like an anvil, on my stomach, and let you hit the anvil with the sledgehammer, it wouldn't hurt me a bit. The inertia of the anvil keps it from moving very much.
  

• Recognize that mass gives an object inertia, a tendency to resist acceleration.
  
• Understand in terms of inertia why heavier objects (car, bowling ball) are harder to move than lighter ones (bicycle, baseball).
  
• Know Newton's first law, that the motion of an object changes only if there is a force exerted on it.
  

1.3. Newton's second law tells us how force, mass, and acceleration are related

• Know Newton's second law: F = ma
  
• When looking at an equation, be able to say how a change in one variable will change another variable
  
• Recognize newton's second law as an expression, mathematically, of what we know about inertia
  

1.4. If an object is not accelerating, the forces on it in each direction balance

1.5. An object's weight is the force exerted on it by gravity

1. All objects in free fall are being acted on only by gravity, and that
2. All such objects accelerate at a rate g = -9.8 m/s² that says how strong gravity is on Earth, and that
3. The force acting on an object is its mass times the acceleration (F = ma), so it should come as no surprise that

1.6. An object resting on a surface is supported by a "normal" force balancing its weight

• Draw force diagrams with forces in the horizontal and vertical directions.
  
• Determine the net force from a force diagram.
  
• Determine the missing forces in a force diagram, given the net force.
  
• Recognize that the net force is what is used in Newton's second law.
  
• Recognize that no acceleration means a net force of zero.
  

1.7. If I try to slide an object along a surface, a friction force resists this

The equation relating all these things is shown off to the left: the Force of friction between two surfaces is their μ multiplied by the normal force pressing them together.

Really, however, this only gives a maximum value for μ. Like the normal force, friction is lazy; if it can exert less than that and still keep the object from moving, it will. Unlike the normal force, friction hits a limit where it can't push any more, and it will give up and let the object start moving.

So, for example, in the situation above, I am pushing a crate which has a mass of 10 kg, and therefore a weight of 100 N. The normal force will be exactly strong enough to keep the crate from falling through the ground, so it will be exactly 100 N, to balance gravity. The μ of the crate is .5, so the maximum strenght of the friction force is μF_N = 50 N.

I am only exerting 49 N of "applied" force toward the right. Normally, this would be a net force, and the crate would accelerate; in this situation, however, friction is present, so it will try to oppose that motion by pushign off to the left. Friction can grow as big as 50 N, but in this case only 49 N is needed to oppose my force and keep the crate from moving, so friction, being lazy, will be exactly that strong.

Now suppose a mouse were to come along and push with an additional force of 2 N, so that together we are exerting 51 N of force on the crate. Now friction is unable to cope, and it will max out at 50 N, leaving one newton of force to accelerate the crate to the left (albeit very slowly).

Often it happens that the friction force actually becomes less once the object starts moving, because it gets somewhat stuck in place when it stands still and once that initial resistance is overcome, sliding it along becomes easier. In a situation like this, we say that the surface has two different coefficients of friction: μ_s, where the "s" stands for "static", for objects that are not moving, and μ_k (for "kinetic") when the object is moving.

The difference between static and kinetic friction is why you will often lurch forward briefly when you first get an object moving. It happens because when the object stands still, it has time to form bonds with the surface it is on. The static friction will always be more than the kinetic friction.

Material
glass-glass	0.9	0.4
metal-metal	.6	.4
greased metal	.1	.05

So, for example, when two glass surfaces are in contact, there is coefficient of friction of .9, almost as strong as is possible. Once you get them to start sliding, however, this coefficient will drop to .4, meaning that the friction force will be very markedly less. A similar, but less dramatic thing happens when metal is in contact with metal. If the point of contact is lubricated, both the static and kinetic friction are reduced dramatically; this is why any machine with moving parts needs to be oiled or greased.

Activity: Friction Lab (pdf) (10 pts, 11/13/06)
(Find the μ, the coefficient of friction, for a block sliding along a table, by weighing it down with different amounts of weight and determining the normal and friction forces)


Homework: Friction (pdf) (Due: 11/14/06)

• Be familiar with the idea of μ, the coefficient of friction, representing how sticky a surface is.
  
• Know how μ relates the friction force and the normal force.
  
• Recognize that pressing two surfaces together more increases the force required to slide them past each other, and that the normal force is the force with which two surfaces are pushed together.
  
• Be able to solve out force diagrams involving normal, gravity, friction, and applied forces, acting in either the vertical or the horizontal direction.
  

2. A force at an angle can be "resolved" into components using trig

2.1. The trigonometric functions relate a triangle's side lengths to its angles

We know now that the forces acting on an object are all vectors, and that when added as vectors, the result is the net force. So, if we can measure the strengh and direction of the forces acting on any object, we should be able to draw a force diagram and find the net force.

Once we start using vectors in "real life" problems, you will find that we rarely know the + form of a vector. Instead, we typically know the length of the vector, and what direction it points in. So for example, in a plane, I can know that I am going NE at 71 m/s; I am unlikey to have a speedometer that reads this out as (50 + 50) m/s.

So, the problem we are trying to deal with is basically this: We know the length of a vector, shown here as the hypotenuse of a right triangle, and the angles of the triangle. We know from geometry that if we know any side of a triangle and we know the angles of the triangle, there is only one possible shape that that triangle can have. Therefore, there must be some way to figure out the length of the sides, the and components of .

In order to do this, we will use the field of mathematics called trigonometry. Trigonometry, as the name implies, is about measuring the sides and angles of a triangle. Of particular interest to us are three functions that your calculator knows how to calculate, called sin(), cos(), and tan(). These functions allow us to find out the ratio between any two sides in a right triangle, given that we know the angles of the triangle.

If we consider an angle a in a right triangle, and label the three sides as the Hypotenuse, the side Opposite the angle a, and the side Adjacent to it, then:

• sin(a) gives the ratio Opposite / Hypotenuse
  
• cos(a) gives the ratio Adjacent / Hypotenuse
  
• tan(a) gives the ratio Opposite / Adjacent
  

You can remember these ratios with the mneumonic SOH-CAH-TOA.

So how do I use these ratios? Suppose that I know that = 6 N, and that this force is at an angle of 30°. In order to add it to other forces, I have to find its and component form. So, I have a triangle where the Hypotenuse is 6 long, one angle is 30°, and I want to find the y, which is Opposite that angle, and the x, which is Adjacent to it. I simply write down the facts that I know about the trig ratios:

So, my force vector is = (5.2 + 3) N. As you can see, all that I need to do to solve for any side in the triangle, given a side I know, is to write an equation representing the fact that "sine is the ratio of opposite to hypotenuse" or whatever, and solve that equation.

In practice, you can always get away without using tan(), so if you want, you can just forget for now that it exists.
Homework: Trigonometry (pdf) (Due: 11/16/06)

• Recognize that the trig functions give a ratio between two sides of a right triangle.
  
• Recognize that the angles of a triangle sum to 180°, so that if one is a right angle the other two must add up to 90°.
  
• Be able to solve for any side of a right triangle given one side and one of the acute angles.
  

2.2. Trig can be used to take a force at an angle and "resolve" it into components

In a lab, you can measure the strength of a force easily enough. For example, a spring scale connected to a string tells you the amount of tension force in that string. What is a bit more tricky is to measure the angle of the force; this is tricky because we have to measure the angle relative to the horizontal or vertical direction, not relative to any string.

In order to measure the angle of a rope or string, we use a protractor with, attached to the center of it, a weight on a string (this is technically called a "plumb bob"). The weight will hang straight down. Therefore, if I line up the bottom of the protractor with a string, the angle measured on the protractor is the angle between the string and the vertical direction.

So, for example, suppose that I have a string with a spring scale on it, and the scale is reading 40 N. The string goes off toward the left an up, as shown to the right, so I know that the is positive and the is negative. I will measure the angle by holding a protractor near the string; this tells me that the angle between the force and vertical is 60°.

I draw a triangle of dotted lines, showing the amd components. I draw this triangle above the vector, so that it will include the angle I just found. Then, I just need to do a bit of trig to find the missing sides:

So, my force vector is = (-34.6 + 20) N. Notice that I have made the negative, because the vector goes toward the left.

If there were other forces acting on the object, I could do the same procedure to resolve them into components. Then, once everything is in component form, I can breathe a sigh of relief and stop worrying about trig and angles, because component-form vectors are much easier to deal with. In the above problem, for example, if the only other forces acting on the object were a horizontal force from another string and a downward force from gravity, then I could say with certainty that the gravity force is = -20 N (a mass of 2 kg) and that = 34.6 N is the other tension force.

Activity: Components Lab (pdf) (10 pts, 11/15/06)
(Three strings are all tied to a button, then each is connected to a spring scale. If you hold one string horizontal, one vertical, and the other an an angle, you should see that the forces on the horizontal and vertical scales are the and components, respectively, of the angled force.)


Homework: Resolving Components (pdf) (Due: 11/17/06)

• Given an angle, construct a useful triangle on which to apply trig.
  
• Given a force and the angle at which it is pulling, "resolve" it into and components.
  
• Find the net force when several angled forces are added.
  

2.3. Force needs to balance in both components for an object to not accelerate

Suppose that I know that all the forces on an object are balanced, because it is not accelerating. If that is true, all the vectors of the forces mut add to zero; in both the and the direction, the forces balance.

In a situation where we know the forces will balance, we don't have to be restricted to just finding forces, resolving into components, and adding up to find the net force. Instead, as we did with force diagrams before, we can solve to find any unknown force.

So, for example, consider the situation above. I have a wire stretched between two trees, with a spring scale near one end of the wire. A bird lands on the wire, causing it to bend slightly. I can measure the angle of the wire where it touches each tree, and I can see that the force in the upper part of the wire is 50 N. I want to find the mass of the bird.

I can quickly use trigonometry to find out the the upper part of the wire exerts a tension force of = (-43.3 + 25). I know that is straight down, only in the component. So, the component I just found must be balanced by the of the other force; it should be 43.3.

Now, I can use trig ratios again to find out how strong the other force is, and then what its y component is:

So, my other force vector is = (43.3 - 11.6) N. Assuming that the bird is not accelerating, I know that all the forces should add up to zero:

+ + = 0
-43.3 + 25 + 43.3 - 11.6 + = 0
14.4 + = 0
= -14.4

The bird has a weight of 14.4 N, which means it has a mass of about 14.4 / 9.8 = 1.47 kg, a very heavy bird.

Activity: Tension Lab (pdf) (10 pts, 11/16/06)
(Given a situation in which a weight is suspended by strings at many different angles, determine the angle of each string, then rsolve each force into components, and verify that the forces balance)


Homework: Balancing Forces 1 (pdf) (Due: 11/20/06)

• Given a force diagram with many forces, some of them at an angle, determine the net force, or the strength of any missing force.
  
• Recognize that any part of an object has its own force diagram, and must balance individually.
  
• Be aware of the ways that friction and normal forces are deceptively intertwined.
  

One situation that is complicated enough to bear special attention is what happens to the friction force when there is some angled tension pulling on an object. You should remember that the friction force has strength F_F = μF_n. We have talked much about how differences in the surface changes the stickiness μ. It also changes the friction force if I change the , by pushing down or pulling up on the object.

So, for example, consider the situation above, where the man is tring to pull the dog forward with an angled leash. The vertical part of the leash force, plus the normal force, will balance gravity, the only downward force. So, the normal force is less than what it would be if the dog were just standing there, and thus the friction force is less as well.

This doesn't mean the person will have any easier a time of it, however. He is now not just exerting enough force to pull the dog forward, but also is having to pull the dog partway off the ground. A lot of the force he is exerting is "wasted" on just lifting. This is one reason why the dog lowers its head to resist better (the other reason is about leverage: its front feet will be planted more effectively if they are roughly in line with where the force is coming from, just as, when you want to push a heavy box, it is easier to crouch down and push it from near the middle rather than from the top).

Activity: Friction Lab 2 (pdf) (10 pts, 11/20/06)
(One of the hardest things to understand in a force diagram is that the normal force does not always equal the gravity force. If I am pulling up on an object, the normal force will be lessened, until I finally take on all the weight of the object and it lifts off the ground. Similarly, if I am pressing down on an object, its normal force grows, since the normal force must balance my applied force as well as the gravity force)


Homework: Balancing Forces 2 (pdf) (Due: 11/21/06)

• Be able to find the strength of a force, given one of its components.
  
• Be able to find the net force from acceleration and use that to solve a force diagram.
  

2.4. Review of the force types we know so far

At this point, you should be able to look at any situation, identify what forces might be acting on an object, and do what it takes to measure those forces. You should also know how to find some values related to forces:

1. The mass of any object, related to the weight.
   
2. The acceleration of an object, which is related to net force and mass.
   
3. The coefficient of friction wherever there is a friction force.
   

There are five different force types that you need to be able to think about. Each one has very specific rules that determine how strong it is and in what direction it points. For example, tension, normal, and friction forces will all grow larger to try to oppose any other forces, but friction has a limit to how big it can become.

Force	Symbol	What is it? When it is there?	Direction is...	Size is...
Weight		Anything on Earth is pulled downwards by Earth's gravity.	Straight down.	The object's weight, which doesn't change.
Normal		Solid objects don't like other solid objects to try to go through them. Any time two objects are in contact, they push against each other.	Perpendicular to the surface.	Enough to balance force into surface.
Tension		Present whenever a rope or string exerts a force on something.	Along the string.	Any strength, same at both ends of rope.
Friction		When two surfaces are pressed against one another, friction resists them trying to slip.	Opposing the force or movement.	μFN (μ is different for every pair of surfaces)
Applied		An "applied force" just means you're pushing something.	Any direction.	Any strength.

Homework: Force Diagrams Review (pdf) (Due: 11/15/06)

• Identify what sort of forces are acting on an object: gravity, normal, tension, friction, applied.
  
• Recognize that forces add as vectors, and find the net force.
  
• Using trigonometry, find a componnt given the full force, or find the force given a component.
  

3. Exerting a force on any object makes it push back on you

3.1. Force diagrams for multiple objects

When we talked about what direction a force acts in, we said things like that a normal force always pushes away from where two objects are in contact, and a tension force always pulls along the rope or string. In both these cases, there are more than one objects involved, and each is experiencing a force.

If there are several objects in a situation, and I want to understand what is happening to each of them, I can draw an independent force diagram for each object. We call this a free body diagram. Take a look at the situation below, where I have draw a free body diagram for each of three blocks that are stacked on top of each other.

Here, the top box experiences just what any box on a table would feel: a gravity force pulling it down, and a normal force from the second box, pushing it up. The second box, however, has the normal force of the top box pushing down on it; this means that, in order for the forces on it to balance, there must be twice as much normal force pushing up from the block under it. We could think of this as being because the bottom box has to support both of the top two. The bottom box, in turn, has the weight of two boxes pressing down on it, in addition to its own weight, so the normal force on it from the table must be three times the weight of a box.

This is a slightly more complicated example. Look through it and try to figure out where all those forces come from.

Three forces that should be easiy to explain are the gravity force on each ball. As we would expect, the biggest ball has the most weight.

We can also see the normal force from the ramp on each ball. This is the one that goes up and to the right. You can check and see that it is perpendicular to the ramp, and that it points in the direction that is opposite where each ball contacts the ramp.

The balls also exert normal forces on each other, just like in the stack example. If you circle the points where each pair of balls touch, you will note that each ball has a vector pointing away from that point.

Finally, the bottom ball has a friction force actng on it. Where does this come from? Friction always has to be along a surface touching the object; the fricton vector will be perpendicular to the normal vector from that surface. In this case, the friction is from the ramp. This friction force is what is responsible for supporting the whole stack of balls.

Notic that the friction force is bigger than the normal force: μ is greater than 1. Maybe the bottom ball is really a wad of chewing gum.

Here's one last situation with its free body diagram. Notice that each ball has its own weight and its own tension force toward the outside, but the tension force toward the inside must be the same, because it's the same rope and it has a certain amount of tension in it. The direction of the two forces are opposite, because tension always points along the rope.

3.2. Newton's Third Law

In the examples above, we see that force really comes from two objects interacting; no force just acts on one object alone. When two objects are touching, each feels an equal normal force pushing them apart. The two objects on either side of a rope are pulled toward each other with equal forces. The friction force that I exert backwards on the groun, when I start running, corresponds to a forward force that gets me moving, and I had better hope that the ground isn't ice, or a loose rug, or something that can't give that much friction. Even the gravity force on me from the Earth is matched by an equal force that I exert upwards on the earth.

So, what we see is that force comes from an interaction between two objects, and both objects will always feel a force. The direction of the force on each object will be the opposite of the other, but the strength of the force will be exactly the same. We can think of this pair of forces as being part of the same event: molecules in the surface resisting being squished (), a rope fighting against stretching (), two surfaces trying to grab old of each other ().

Newton made note of this fact in his third law of motion:

All forces exist as part of a pair of forces acting on two different objects. If the force on A from B is , the force on B from A will be -: a force of the same strength in the opposite direction.

This law is sometimes stated as "For every action, there is an equal and opposite reaction."

Homework: Free Body Diagrams (pdf) (Due: 11/28/06)

• Know and apply Newton's third law.
  
• Be able to sketch the force diagrams for several objects, given the requirement of matching force pairs.
  
• By recognizing force pairs, determine in what way several objects must be in contact with each other.
  

3.3. Reassembling a situation using the force diagram of each object

I particularly enjoy the parts of physics, like looking at a motion graph for multiple objects and determining what story it tells, that involve a bit of detective work and creative thinking. We occasionally have to do a similar sort of thing when dealing with forces: we may be presented with a force diagram for one or several objects, and want to figure out what is happening to those objects.

Activity: Force Puzzles (pdf) (11/28/06)
(Given the free body diagrams for several objects, determine how they must be interacting with each other by using Newton's third law and identifying force pairs)

4. Review

There are three basic questions that you need to be able to answer about Force:

What forces act on this object?

• gravity () is always present.
  
• contact means there is an pushing objects apart.
  
• a rope will create pulling the objects together.
  
• objects sliding past each other feel resisting this.
  

//Put up pictures of the direction of forces
How strong are each of those forces?

You should be able to use the equations that tell you how strong a given force is. In addition, you should be aware that the friction force given by the equation is the maximum amount that friction will push. Being lazy, it will push less than this if that is enough to keep the object from moving. If the object is moving, however, friction always has its maximum value.

What do those forces do to the object?

The rule is that if you find the net force, the sum of all the forces acting on an object, that will tell you how much hte object accelerates. This also works in reverse: if I know that the object does not accelerate, then I know that the net force on it is zero.

Homework: Force Review (pdf) (Due: 11/29/06)

• Know and apply Newton's three laws.
  
• Draw force diagrams representing any given situation.
  
• Use equations to find the acceleration, coefficient of friction, and gravity force.
  
• Balance a force diagram, resolving forces into components as needed.