Forces
At this point, we have talked about some of the math needed for this course, and we have also talked about motion. We will now move to talk about what causes motion, which- simply put- are forces.
force: an influence on an object that causes a change in its motion, shape, or both.
To deal with forces properly, we need to discuss Newton’s Laws of Motion, or what I like to call “the rules of the game”:
I) If the net force on an object is zero, its velocity will remain constant. II) A force exerted on an object will result in an acceleration.
III) A force that one object exerts on a second object, the second object exerts onto the first with equal magnitude and opposite direction.
The first law is the law of equilibrium, a state in which all forces sum to zero. This means that an object either sitting still or moving can be in equilibrium; as long as the net force is zero, an object will continue along with a constant speed in whatever direction its velocity vector points.
If we interpret the second law mathematically, we arrive at the equation: . As with the motion vectors, there are vector hats on the force and acceleration parts of this equation because these can be vectors: they can have a magnitude and direction. The mass part of this equation, however, does not have a vector hat because mass is a “scalar” quantity- it is simply a number. For a refresher on vectors and scalars, refer to that section of these notes.
Before moving farther, we need to make the distinction between mass and weight.
mass: the amount of “stuff” an object is made of.
weight: the force exerted on an object as a result of gravity.
In the definition for mass, “stuff” refers to the total amount of matter (or anti-matter) that an object is made of. It is usually measured in grams, though in physics, we tend to use kilograms
because of the size of the objects we deal with. Because mass is the amount of material that makes up an object, its value does not change when an object moves from one place to another. An object with a mass of on Earth will still have a mass of on Earth’s Moon, or any other moon, for that matter.
Weight, as shown in its definition, is a force: namely, the force due to gravity. To find a value for weight, we multiply an object’s mass by the acceleration due to gravity in its vicinity. So, an object with a mass of on Earth, where the acceleration due to gravity on Earth’s surface
is approximately , would have a weight of . In
Now, what kinds of units do forces have? As shown above, the units work out to , but it can be cumbersome to write these units repeatedly in each term of a given problem. Instead, we can group these terms together under a new label: , for Newtons, after Sir Isaac Newton.
Before going further, it is important to discuss the different types of forces that can influence an object. The origin of these forces depends on how an object is influenced, but the forces we will deal with generally fall into one of these five categories:
I) A push or a pull
a. These influences are fairly straightforward- an arrow detailing their magnitude and direction is all that is needed.
II) Contact, or normal, forces
a. Contact forces arise from the interaction between an object and a surface. For example, a textbook resting on a desk is acted on by gravity. The reason it does not fall through the desk is because the desk supplies a contact force to keep it from moving downward. This force always points perpendicular, or normal, to the surface.
b. Contact forces are not necessarily reaction forces from Newton’s Third Law; the “textbook on a table” example is a consequence of the force of gravity acting upon an object being applied to the desk, but it is not truly a reaction force in the Third Law sense. A better example would be applying a force to a wall; the normal force, in this example, is a reaction to your applied force on the wall.
III) Friction
a. Friction is a force that always points in opposition to apparent motion. For example, in order to walk forward, you must push backward on the floor. The reaction to this is that the floor pushes you, and the reason for this is friction. IV) Tension
a. Tension is a force that acts along cables and wires. We always treat the tension in a cable to be constant throughout its entire length, as treating the problem otherwise would imply that the spring were breaking or failing in some way.
V) Spring
a. Springs exert a force that is proportional to how much they are stretched or compressed. If a spring is compressed, it exerts a force to try to return to equilibrium, and vice versa.
VI) Fluid Drag / Air Resistance
a. Fluid drag is a force that acts on objects that are moving through a fluid. For reasons discussed later, both liquids and gases can be described through the same equations. Generally speaking, the more quickly an object is moving, the more the medium- liquids or gases- will push on the object.
It is also helpful to draw what is called a “force diagram” to visualize the problem we are discussing. In this diagram, we draw all forces acting on the object in question. If we want, we could also draw other vectors, such as a velocity vector, but to keep it simple in this example, we have only drawn forces. This example has several forces acting on it:
In order to find the net force, we can use vector addition (graphically or mathematically) on these forces. Once we have these forces displayed, we use the coordinate system to sum the five forces using vector components, build the final resultant vector, and report the net force on the object, which may look like:
As an example:
Two warehouse workers are in a disagreement over which way they should move a 10 kg box of supplies. Worker A applies a force of 50 N to the right, while Worker B applies a 100 N force to the left. Find the net force acting on the box, and then find the box’s net acceleration.
Draw the diagram:
The most detailed and organized way to go about solving problems like this is to use Newton’s Second Law. In this case:
In this case, the object is not accelerating in the direction, but in principle, it could!
So, then, the net acceleration can be found:
And the net acceleration is found, in general, by using the magnitude of its vector components. In many problems, the net acceleration will be equal in magnitude to one of the components, as we tend to try to orient our coordinate axes such that one of the force equations is set equal to zero. The reason for this is that it makes much of the algebra easier to work through.
It is also important to note that even though the acceleration is negative, as in kinematics, this does not necessarily mean that the object is slowing down! It could be speeding up to the left.
Friction In this section, we discuss a specific force: friction.
friction: a force that opposes apparent motion
The exact nature of friction- where it comes from, that is- still remains unsolved. There are certainly contributions at the atomic level, but how the contributions and properties of the atoms result in what we observe still eludes our attempts to fully explain it. That said, we can still describe friction well enough to suit our purposes.
Start with a question: have you ever noticed that when you push a large object, it is often much easier to keep it in motion than it is to “pop it loose”? Why is this? Let us keep this question in our minds as we go further.
There is a mathematical formulation for friction, and it is:
The Greek letter (pronounced "mu") is the "coefficient of friction." It has no units (as you may be able to see by the formula; the units cancel out), and is defined as the ratio of the frictional force and the normal force. Its value typically ranges from near 0 to 1, though larger coefficients have been measured.1
The actual value of the coefficient of friction depends mostly on the surfaces that are in contact with each other. In reality, the mass of the objects we are considering do play some role, as does
1 There are cases in which there are negative values for coefficients of friction, but these are very exotic and rare,
surface area. However, to simplify our discussion, we will consider the coefficient of friction only changing as a result of the surfaces in contact; the other effects are relatively small, so we will discount them. We do not yet fully understand the nature of friction, but the scientific community is working on more research into the problem.
So, we can start to analyze situations involving friction. This is useful, since we do not live in a frictionless world. There are three types of friction: static (stationary), kinetic (moving), and rolling friction. We use the same formula for all of these, just different numbers are plugged in for the coefficient of friction. The kinetic coefficient of friction is smaller than the static coefficient of friction in all the cases observed thus far, and typically, rolling friction is even smaller. When we analyze this below, we will explore, physically, the reality of the question we asked at the beginning.
Consider an object sitting on a floor, and we exert a force to try to push it. If we draw all forces acting on this object, we get the following diagram:
is the weight, or gravitational force, of the object ( ); is the normal force- or the force exerted by the surface. As discussed previously, this force is perpendicular, or normal, to the surface, and in cases like that seen above, is equal to the weight of the object as a result of Newton's 3rd Law. Then, is the force we are applying; and is the force of static friction, and is given by the formula in the upper-right-hand side of the diagram. For convenience, we can just use the subscript for static friction, or for kinetic friction, or just for friction force, as long as we are clear that this force is the friction force.
The reason the force of static friction is described using an inequality, rather than an equation, is because the maximum force static friction can exert is equal to ; when we apply a steadily increasing force, the force of friction goes from 0 to that maximum, as the force we apply approaches that maximum. We can see this mathematically by trying to find the net force, if we let the force we exert be less than or equal to the maximum. Taking up to be positive in the y-direction and right to be positive in the x-y-direction, we find:
x-direction: y-direction:
Once the force we apply exceeds , then the object begins to move, and we now must have a similar diagram, but with the kinetic friction force being applied:
Here, the arrow in the upper right tells us the object is moving to the right.
Once more, if we find the net force:
x-direction: y-direction:
So, the net force would appear to be to the right, which we might expect- given how we've constructed the diagrams.
Now, where in the static case, if the net force is zero, the object did not move. But in the kinetic case, if the net force is zero, what can we say about the object's velocity?
Tension
Tension is a force exerted by a string. The direction the tension force points is determined solely by the string being used; whatever direction the string is pointing, this is the direction of the tension force.
We also assume, at this level, that strings/cables/etc are massless, as it is difficult to account for the tension force otherwise. As a result of this assumption, we assume that the tension force at every point in the string is equal. (If it was not, this would essentially mean that the string was breaking- in which case, we would no longer have a tension force!)
Springs
He was, however, able to claim several achievements besides his biological discovery. Chief among them, in physics, is his work in the study of springs. He found that the force that springs exert on an object is not constant- in fact, it depends on how strongly a spring is stretched or compressed. He formalized this relationship into Hooke’s Law:
Where is the force that the spring exerts, is the displacement of the spring from its equilibrium state- how much it is stretched or compressed- and is the spring constant. The spring constant is determined by a variety of factors- the material the spring is made of, the length of the spring, and so on. In the “real world,” springs must have their spring constants determined by experiment, though it is possible to manufacture springs with a given spring constant. The negative in this experiment tells us that the force always points toward the
equilibrium position: if a spring is compressed to the left, its restoration force points to the right. And if a spring is stretched to the right, its restoration force points to the left.
Fluid Drag / Air Resistance
Fluid drag, or air resistance, is a force that resists motion. It is tempting, therefore, to call this force “air friction,” but this word is technically incorrect. The reason for this is that, with friction, we describe two surfaces interacting. With fluids, we are describing atom-to-atom interactions, which are not quite like friction.
Air resistance is the source of what is known as terminal velocity- the velocity at which an object will fall without accelerating. We discussed in the kinematics unit how all objects accelerate toward the surface of the Earth at a rate of , but this was in the absence of air
resistance. If we account for this, the acceleration slowly shrinks, because as an object begins to move faster, air resistance pushes with more and more force. This will be true until the force exerted by air resistance is equal to the weight of the object, at which point- by Newton’s First Law- its velocity will remain constant. We call this velocity terminal velocity.
So, in all practical terms, when:
will remain constant.
Systems of Objects
So, everything we have discussed so far works well to describe many combinations and
We will define a system as a collection of two or more objects on which forces act. The rules for solving Newton’s Laws problems with systems are exactly the same as they were before: sum up forces on a particular object. The only difference is that some of the reasoning can get tricky to keep track of.
For example: let us start with two blocks of mass and resting on a flat surface, as shown below. A string connects them both, and then the rope is pulled to the right with a force . If we assume that there is no friction between the blocks and the surface, what is the acceleration of the system?
Before, we would just take the force and divide by the mass of the object, from Newton’s Second Law. Can we do the same, here? It is not so clear. Let us, instead, look at each object separately, and draw force diagrams for each:
Motion on Inclined Planes
Before we do problems, we need to do one last thing with motion, which is deal with inclines:
+y
Here, the weight is still straight down, but the normal force is off at an angle. You may also notice our coordinate axis is tilted at an angle, such that the x-axis is along the incline, and the y-axis is perpendicular to the incline. We can break the weight force into components along these
axes; using a geometric proof of "mutually perpendicular sides," the formulas we use involve the same angle of the incline:
x-direction: y-direction:
And the result looks like:
+y
If we do not have friction on this incline, then, the motion of objects on an incline can be discussed in terms of acceleration due to a fraction of gravity, gravity, dependent on the angle by that equation: . When , we have a flat surface, and acceleration in the
horizontal direction due to gravity . As we increase the angle, this acceleration in the horizontal direction on our coordinate axis will increase until it reaches a maximum of at
.
We can consider the case of an object sliding along a frictionless surface toward an incline; it will move with constant velocity until it encounters the incline. Once it begins sliding up the frictionless incline, the basic equations of motion we’ve become familiar with will govern its
motion, with no needed modification.
So, suppose we have an object resting on an incline. If we take friction into account, what does the force diagram look like?
Consider the object sliding up an incline; which way would friction point? Friction opposes apparent motion, so friction would actually point down the incline. As a result, the net acceleration would be a component of the object’s weight plus the acceleration due to friction. The basic equations of motion would still apply, we would just need to modify our acceleration
term.
Here's an interesting thought: we may recall from common experience that if we lift something up at a high enough angle- say, a desk with a book on it, or a book with a pencil on it- the book will begin to slide, because its own weight has exceeded the frictional force supplied by the
incline. We can find that angle using algebra!
If the object is just about to slide, the component of the weight down the plane must equal the force of friction:
Solving for the coefficient of friction:
This angle is called the critical angle of static friction, or the angle at which static friction is overcome by an object’s own weight.
This derivation tells us that the angle at which the object slides is also independent of the object's mass; it only has to do with the surfaces that are in contact with each other. (Try this at home- if you have two blocks of wood, put one on the other, tilt them until the one on top slides, measure
Now, let us make things more interesting. Suppose that we do not shift our coordinate axes with the table, instead keeping them fixed in their usual orientation. Would that change the above
derivation? Let us see!
Instead of having forces line up very helpfully, we will now have different components of each force. So:
If the block is at rest, each acceleration is zero, so:
From the first equation, we find that:
We can use the second equation to actually find the magnitude of the normal force in this situation:
And we can notice that this is true only so long as the denominator is greater than 0. If it were less than zero, this would imply that the normal force was pointing down, which does not make
sense. It also cannot equal zero, as that would make the normal force undefined. So, we find a limiting condition:
