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Newton's Laws

Sir Isaac Newton made numerous and important contributions to the study of physics, particularly in the use of calculus. Among his contributions are his three laws relating to forces and motion. In this lesson, we will cover Newton's three laws of motion, but first we must begin by studying some of the basic properties of force.

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Basics of Forces

  • A force is, put in simplest terms, a push or a pull. For example, if you push a stack of books across a desk, you are exerting a force on the stack of books.
  • A force must be exerted on one object by another. Using the previous example, the force is being exerted on the stack of books by you.
  • When we talk about forces, in most cases it is numerically unimportant what the force is exerted by. For calculations, we are only interested in what the force is exerted on.
  • Though a force usually causes the object it is acting on to move, this is not always true. For example, if you tried to move a sleeping elephant by pushing on it, the elephant wouldn't move an inch. However, you would still be applying a force to the elephant. We will return to this example later to explain why the elephant does not move even though you exert a force on it.
  • Force is a vector quantity, meaning it has both a magnitude and a direction. Forces are usually denoted with a bold, uppercase letter F, and often have subscripts to denote what is causing the force. Force is measured in Newtons, abbreviated N.
  • There are two types of forces: contact forces and field forces.
    • Most forces are contact forces, which occur between two objects that are in contact with each other. All of the examples we have used so far involved contact forces.
    • Note that contact forces do not have to be exerted through direct contact. They can instead be exerted through indirect contact, such as exerting a force on an object by pulling on a rope attached to it. Contact forces can even act through the air, such as when you blow on a piece of paper. You exert a force on the air, which in-turn exerts a force on the paper.
    • Field forces are fewer in number, but no less important. Field forces act whether there is any contact between the two objects or not. A field force is exerted by one object on all other objects within a certain distance, the magnitude of the force decreasing as the distance increases.
    • Theoretically, the radius in which a field force acts is infinite, but beyond a certain distance it becomes so small that it is completely negligible.
    • The most common field force is the gravitational force, which will be discussed later in this lesson and in more detail in the lesson on gravitation. Other field forces include the electrostatic force and the magnetic force, which will be discussed later in electricity and magnetism.

Newton's Laws of Motion

  • Newton's First Law
    • Newton's First Law states that an object in motion will remain in motion, and an object at rest will remain at rest, until a force acts upon it. This is often called the Law of Inertia, and an object's tendency to follow this rule is often called its inertia.
    • Another way of stating this law is that an object's velocity will remain constant until a force acts on it. For example, a book sitting on a desk will never move unless someone applies a force to it, and a ball rolling at a constant velocity will continue to roll at a constant velocity until a force acts on it, either increasing or decreasing its velocity.
    • You may be thinking something like "but wait, a ball doesn't roll at constant velocity, it gradually slows down until it stops." This is true, and the reason for this is that a force is acting on it: friction. Friction acts against the ball's motion, causing it to slow down. We will learn more about friction later in this lesson
    • This brings us to another important point: equilibrium. When we talk about forces, equilibrium is the state in which the sum of all forces acting on an object, or the net force Fnet, is zero. For example, when you press the gas pedal on a car, a chain of events occurs which results in a force that pushes the car forward. Friction applies a force that acts opposite the car's motion. These forces act in opposite directions, so if they have equal magnitudes, then they cancel each other out, resulting in a constant velocity.
    • There is also a quantity that can be thought of as the measure of an object's inertia. Mass, abbreviated m, is a scalar quantity that measures how much matter is in an object. Mass is measured in kilograms, kg, and it is important to note that mass is not the same as weight, which will be discussed later in this lesson.
    • The more mass an object has, the greater its inertia is, and thus the greater the force required to cause the object's motion to change. This is part of the reason why trying to push a sleeping elephant in our earlier example is so difficult: the elephant is very large and has a lot of mass, so it has a large amount of inertia, meaning a large force is required to cause it to move.
  • Newton's Second Law
    • Newton's Second Law states that The net force on an object is proportional to the mass and acceleration of the object. Acceleration is directly proportional to the net force applied to the object and inversely proportional to the mass of the object.
    • This can be expressed mathematically as Fnet = SF = ma. S is the Greek letter Sigma, which is used in science to denote sums. So, SF is the sum of all forces acting on an object. This equation is contended by some sources to be the most important equation in mechanics, and no doubt it should be learned very thuroughly.
    • From this equation, we see that 1 N = 1 kg*m/s2. Manipulation of this definition of the Newton may be required in some instances in order to determine the relationship between two quantities.
    • From this equation, we also see that for a given value of net force, increasing the mass will decrease the acceleration, and vice-versa. Going back to our earlier example of trying to move a sleeping elephant, this is one reason the elephant doesn't move. The elephant's mass is so large that the force a person is physically able to apply produces only a very tiny acceleration, which is canceled out by friction.
    • Note that since Fnet = ma, and m is a scalar, the direction of Fnet is the same as the direction of a. Also note that this is vector multiplication, meaning that directions must be taken into account. Because of this, it is common to use two net forces, one for the x direction and one for the y direction.
  • Newton's Third Law
    • You have most likely already heard Newton's Third Law, stated as for every action, there is an equal and opposite reaction.
    • What this really means is that when Object 1 exerts a force on Object 2, Object 2 always exerts a force on Object 1 that is equal in magnitude but opposite in direction. These two forces are often denoted F1-on-2 and F2-on-1, and are called an action/reaction pair.
    • This is often misread to mean that the net force on the objects is 0 because of the equal and opposite forces. For example, if this were true, hammering a nail would be impossible because the force exerted on the nail by the hammer would be balanced out by the force exerted by the nail. However, the reason this is not true is because the forces are not acting on the same object, only one of these forces is acting on each object. So the nail is accelerated by the force exerted by the hammer, and the hammer is accelerated back up by the force exerted by the nail.
    • This is also the principle behind all rockets. Taking a water-fueled bottle rocket for example, the water is pressurized so that when it is released, it pushes out toward the ground with a large force. When the water hits the ground, it exerts that force on the ground, which by Newton's Third Law exerts a force of equal magnitude back up on the water. The water then transfers this force to the bottle, causing it to accelerate upwards.
    • Something that you may have noticed in this rocket example is that, though the force exerted by the ground caused the rocket to accelerate and shoot up, the ground did not accelerate under the same force. But this is not entirely accurate: the ground and actually the entire Earth did accelerate. However, because the Earth's mass is so overwhelmingly enormous, the force produced such a small acceleration that it was completely negligible. This is the case anytime a force is applied to the Earth, such as landing after jumping, dropping an object on the ground, or even doing push-ups.

Everyday Forces

  • Weight
    • Unlike mass, weight is actually a force. More specifically, weight is the force on an object due to gravity. Weight is commonly noted Fg, but may be written as FW.
    • Since we know that F = ma, and acceleration due to gravity is represented by g, we can say that Fg = mg.
    • Note that the acceleration due to gravity is dependant upon the mass of the body producing that gravitational field. So, since the moon is much less massive than Earth, g is much smaller on the moon than on Earth, and as a result everything weighs less on the moon. However, you should also note that the mass of an object is the same whether it is on Earth, the moon, or anywhere else. Unless stated otherwise, all problems occur near the surface of Earth, where g = 9.81 m/s2.
  • The Normal Force
    • When an object, such as a stack of books, sits on a surface, such as a desk, gravity is applying a downward force on that object, trying to make it move downward. But, a book sitting on a desk does not move downward because it is sitting on a desk. This tells us that there must be some force counteracting the force of gravity, pushing upward on the book. We call this force the normal force, FN or sometimes just N.
    • The normal force exists because of Newton's Third Law. The book sitting on the desk applies the force of its weight to the desk, so the desk applies an equal and opposite force to the book.
    • When an object sits on a flat surface and the only forces acting in the vertical direction are the gravitational force and the normal force, the magnitudes of the gravitational force and normal force are equal. This is true whether the object is moving in the horizontal direction or not.
    • If an object on a flat surface experiences additional forces in the vertical direction but does not leave the surface, then the normal force changes to maintain vertical equilibrium. So if you push down on the book, adding an additional downward force, then the normal force increases to equal the magnitude of the gravitational force plus the force you apply. If you pull upward on the book without lifting it off of the desk, the normal force decreases so that the magnitude of the gravitational force equals the normal force plus the applied force.
    • Note that the normal force is only present when the object sits on a surface. The object can move along the surface in any way without affecting the normal force, but if it leaves the surface, the normal force is no longer present.
    • The normal force also occurs on an inclined surface, like a ramp. However, while the gravitational force continues to point straight downward toward the center of the Earth, the normal force points perpendicular (at a right angle) to the surface. Because of this, an object on an incline has a normal force whose magnitude is less than the magnitude of the gravitational force. However, the magnitude of the normal force is equal to the magnitude of the component of the gravitational force that acts parallel to the normal force, which is usually Fgcosθ.
  • Friction
    • Friction is the force that acts opposite the direction of motion, and is caused by contact between two surfaces. In reality, friction also occurs between any moving object and the air, in which case it is called air resistance or drag, but for the AP and AICE curriculums, this can be considered negligible and ignored.
    • The force of friction, Ff, depends partially on the types of surfaces involved. For example, contact between polished wood and cardboard causes less friction than contact between concrete and sandpaper. The property of the surfaces that affects this is how rough or smooth the surface is. Rough surfaces like unpolished rock produce a lot of friction, but simply polishing that rock would smooth out the surface and lower the friction it produces.
    • In reality, though one can greatly reduce the friction caused by a surface through polishing and using liquids (which generally produce less friction than solids), it is impossible to create a completely frictionless surface. However, some problems you will work with will assume negligible friction for the sake of simplicity, but many problems do require friction to be taken into account. Be sure to look for clues that may indicate whether or not friction applies to the problem.
    • In order to take into account the differences in the frictional force caused by the differences in surfaces, we use a quantity called the coefficient of friction, represented by the Greek lowercase letter mu, μ. The coefficient of friction varies for every combination of two surfaces in contact with each other, and each pair of surfaces has two coefficients of friction: the coefficient of kinetic friction, μk, and the coefficient of static friction μs. You will not be asked to memorize these, they will either be given, or you will be asked to calculate them.
    • There are two types of friction: kinetic and static. Kinetic friction occurs when an object is moving, and acts parallel to the surface (perpendicular to the normal force) and opposite the direction of motion. Static friction occurs when a force attempts to move an object, but that object has not yet began moving. Static friction acts parallel to the surface and opposite the direction of the force attempting to cause motion. You may have noticed that it is more difficult to get a heavy object to start moving than it is to keep it moving. This is because the maximum force of static friction is greater than that of kinetic friction.
    • Now that we know the difference between kinetic and static friction, we can define a relationship between frictional forces and their coefficients of friction. The force of kinetic friction is denoted Fk. As far as static friction goes, we are really only concerned with the maximum possible force of static friction, Fs,max, which is equivalent to the maximum force that can be applied to the object without causing it to move. The formulas for the force of each friction in relation to their coefficients are listed below:
      • Fk = μkFN
      • Fs,max = μsFN
    • So, we see that the force of friction is dependant not only on the coefficient of friction between the surface and the object, but also on the normal force. Since the normal force is dependant upon the object's mass, the force of friction is greater for an object with larger mass.

Free-Body Diagrams

  • A free-body diagram is a sketch used to show the forces acting on an object. It is strongly advised that you always draw a free-body diagram anytime you are dealing with forces, because the diagram makes it much easier to solve force problems.
  • Free-body diagrams should focus only on objects that we need to find the forces on. Other objects like ramps can be drawn, but only to show that they affect the object's motion. Only the object of focus is to have force vectors drawn on it.
  • Also, force diagrams should not have any vectors that do not represent forces. So, things like velocity or displacement vectors should be left off of a free-body diagram.
  • Components of forces that are not completely on the axis can also be drawn on a free-body diagram, and this is often helpful. However, it is important to note that for any free response problem on the AP exam, you will lose points if you have both a vector and its components. You must have either the vector or its components though, and since components are generally more useful, it is often advisable to draw the components and not the original vector, unless the components will overlap other force vectors.
  • Finally, one last note before we get into the specifics of drawing free-body diagrams: you must label every vector on a free-body diagram. Not only will you lose points for unmarked vectors, but it will also be more difficult to differentiate between them.
  • Drawing Free-Body Diagrams
    1. First, start by drawing the object of focus and any other important objects. If a ramp is involved, it should always be drawn and labeled with an angle to the ground. Sometimes there will be a system of objects connected in some way, usually either through a rope and pulley or hanged from different spots on a rotating bar (this will occur later in circular motion and rotation). In most cases, strings and ropes should not be drawn because they can cause confusion with the vectors along them, but this is more a matter of personal preference. The object of focus should be a simple shape like a square, rectangle, or circle. Do not waste time making fancy drawings.
    2. Next, unless you are dealing with torque (which will not be until circular motion and rotation), draw a dot at the center of the object. All force vectors will originate from this dot for purposes of simplification. The only exception is forces that cause torque on a rotating object, which we will cover in a later lesson.
    3. Unless the object is in deep space far away from any other objects with mass, the gravitational force will always be present on the object. So, start by drawing a vector pointing straight down and labeling it Fg. Vectors are drawn as arrows, and this arrow should begin at the center of the object and point straight down.
    4. If the object is sitting on a surface, there will be a normal force. So, draw a vector for the normal force starting at the center of the object and pointing perpendicular to the surface, and label it FN. Keep in mind that the length of a vector should be indicitive of its magnitude, so try to draw higher magnitude vectors longer than lower magnitude vectors.
    5. Now, draw any forces that are being applied to the object. This includes not only physically pushing or pulling on the object, but also tension forces caused by wires or ropes. For example, an object hanging from a string experiences a tension force along the string. Any tension force should be labeled FT (or sometimes just T), while other forces should be labeled appropriately. Often it is best to use just F if there is only one applied force, and add appropriate subscripts if there are others, or if the force is given a subscript in the problem.
    6. Next, if the object is on a surface that is not frictionless, and there is at least one force that is not completely perpendicular to the surface, draw a vector for the force of friction parallel to the surface and in the direction opposite the way the object would move based on the other forces. This vector can be labeled Ff for any frictional force, Fk for kinetic friction, or Fs for static friction.
    7. Finally, define a coordinate system. Usually the coordinate system will have the horizontal or x-axis parallel to the surface and the vertical or y-axis perpendicular to the surface. Unless the object is on a ramp, the coordinate grid will usually be perfectly horizontal and vertical as normal. However, when the object is on a ramp, it is usually best to tilt the coordinate grid so that the y-axis is perpendicular to the surface, as this will usually give the greatest number of vectors aligned with the axes.
  • Below are some examples of free-body diagrams for some of the most common situations you will see:

Pulleys

  • Most likely you already know what a pulley is: A wheel with some sort of rope over it. Pulleys are used to change the direction of a force, such as being able to pull down on the rope to pull an object attached to the other end upward.
  • For the AP and AICE exams, pulleys will generally be assumed to have negligible friction and mass so that the magnitudes of forces are not changed, only their directions.
  • The most common pulley problems on the AP exam will give a diagram similar to the one below:
  • As you can see, this diagram shows two blocks of masses m and M, attached by a string that goes over a pulley, with the block of mass M hanging in the air. Normally, you will be told that the surface is frictionless.
  • To solve this problem, you should first draw a free-body diagram showing only the two blocks and the forces acting on them. The block of mass m should show the force of gravity, the normal force, and a tension force caused by the string. The block of mass M should have only the force of gravity and the tension force.
  • The problem may ask you for the acceleration of the blocks when they are released from rest, which can be found by writing Newton's second law twice, once for the horizontal direction of the block of mass m and once for the vertical direction of the block of mass M, which leaves you with two unknown quantities: FT and a (note that a will be positive for block m's equation and negative for block M's equation, due to the directions of motion).
  • The next step would be to remove FT from both equations, which can be done by first multiplying block M's equation by -1 so both accelerations are positive, then adding the two equations, which will now cause FT to cancel out of both equations, then solve for a.
  • Problems like this may be given with or without values for m and M. Either way, it is advised to manipulate the equations entirely before plugging in numbers if they are given.
  • If you are given a coefficient of friction between the block of mass m and the table, then you must factor in the force of friction as well, and later plug in Ff = μFN = μmg.

Note that some books may also include uniform circular motion and even torque in this section. However, we will cover these concepts later in the lesson on circular motion and rotation.

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