Goals: The student will learn|
--The formal definition of Newton's 3rd law: "forces always originate in pairs, equal in magnitude and opposite in direction."
--The informal, qualitative version: "Each action has an equal and opposite reaction."
--Qualitative examples, also given in section #18a of "Stargazers":
--recoil of a gun
--Rotation of a sprinkler
-- The reason it may take 3 strong fire fighters to hold and aim a big hose (it kicks back!)
--The reason it is unwise to jump from boat to shore before tying the boat up (the reaction to the force of your leg muscles pushing you forward will push the boat away from shore!).
Why you cannot balance a stopped bike by shifting your weight.
Optional: how one can formally avoid the notion of force in formulating Newton's 2nd and 3rd laws.
Starting the lesson
Start by making clear the different nature and application of Newton's 2nd and 3rd laws (the students might copy from the blackboard some or all the words below):
Newton's 2nd law describes what happens when a force is applied to an object, but says nothing about the way such a force originates.
Newton's 3rd law discusses the origin of forces: they are always created in pairs, equal in magnitude and opposite in direction. It always involves more than one object.
Terms: Newton's 3rd law, recoil.
Guiding questions and additional tidbits:
--What does Newton's 3rd law state?
"Forces always originate in pairs, equal in magnitude and opposite in direction"
If the student answers "Every action has an equal and opposite reaction", ask "what do you mean by action?"
The student should explain it is something like "force" (which in its turn may be described as "the cause of motion, expressed quantitatively")
--Wouldn't a pair of equal and opposite forces cancel each other?
They might if they acted on the same object, but Newton's 3rd law deals with forces on two different objects.
It states that if one object--let's call it "A"--pushes another one--say "B"--with force F, then "B" pushes "A" back with force –F. The two forces have the same strength, but the negative sign shows one has the opposite direction of the other.
(Do not go past here until everyone in class understands this idea!)
We will go into the mathematics later, but first, some examples which make the concept clear.
Can anyone give an example?
Examples should involve forces that actually cause motion.
(If you stand on the floor, but do not fall into the basement because the floor pushes you back with the same force but in opposite direction--this is just equilibrium between opposing forces. Maybe formally it also fits the 3rd law, but the more relevant applications involve objects which accelerate and move.)
(Examples below are most effective when given by students. Examples not given by any student may be presented by the teacher, but in either case, the student should be asked to identify the two opposing forces. [The teacher may also add comments.]
--Recoil of a gun. When you shoot a bullet forward, the gun applies to it a force. Therefore an equal but opposite force must push the gun backwards
[Comment: If you have seen an old cannon from the US war of independence or the US civil war, you know it had a long trace hanging down on the side opposite to the barrel. This was used to hitch the horses which pulled the cannon. But in addition, the end of the cannon also had a place to drive a long spike into the ground, which anchored the cannon and stoped the recoil from rolling it back. Instead, the recoil would make the wheels of the cannon jump up a short distance, with the spike as pivot--then fall down again, at the same spot as before.]
-- A rising rocket. One force is the pressure pushing a fast jet of burning gas out of the back of the rocket. This leads to an equal but opposing force, the thrust on the rocket itself.
[A jet plane operates the same way. Even the propeller of an airplane or ship pushes the vehicle forward, because at the same time it pushes air or water backwards.]
[Around 1900 a popular notion arose that rockets could not travel in empty space "because they lacked air to push against." Dr. Robert Goddard finally demonstrated experimentally that this was false. Actually, a rocket flies better in empty space, since it has no air resistance to overcome.]
-- The rotation of a garden sprinkler--the kind which has 2-3 arms pivoted at the middle, with each arm bending near its end in the direction opposite to that of the rotation. Draw on the blackboard. How does this work?
The end of each arm shoots out a jet of water, and doing so means that it applies a force to the jet. It therefore experiences an equal force pushing it in the direction opposite to that of the jet. Because the jet comes out at an angle to the radial part of the arm, the reaction force makes an angle, too, and that makes the arm rotate, too.
["Surely you are Joking, Mr. Feynman" is a collection of funny stories from the life of Dr. Richard Feynman, an extremely successful physicist as well as an unconventional thinker in many fields. One story tells how in college, he and his fellow students argued, what would happen if one reversed the process--put the sprinkler inside a tank of water, and sucked water in.
They never managed to do it, or to agree on predicting the result. Later others tried it--the process is complicated by other factors and the rotation is either absent or very slow.]
--A big fire hose always has long handles on both sides of the nozzle. Why?
Firemen will tell you the nozzle is pushed back with great force, and it may take several people to handle it. The water jet is thrown forward, so an equal and opposite force pushes the nozzle back.
--When jumping from a boat to shore, it is always advisable to tie up the boat before jumping. Why?
If you don't, as your legs accelerate your body towards shore, they apply an equal and opposite force to the boat, pushing it away.
[If you have no place to tie up, allow extra distance for your jump, an most important--carry the end of a rope, so you can pull the boat back!]
--You sit on a bike which is not moving, and it starts to fall towards the left. Can you balance it leaning your body to the right?
No! Pushing your body to the right creates an equal and opposite push on the bike to the left, making the problem worse.
[Trained circus performers can balance a stationary bike, but theirs is a complicated and difficult technique.
Riders on a moving bike balance it by turning the front wheel left or right. Because of conservation of angular momentum, such motions rotate the entire bike and its rider, around the line on which the wheels touch the ground, and such rotation can straighten-up the bike.]
(End of examples)
--Newton's 3rd law speaks of "equal and opposite forces." The meaning of "opposite directions" is clear--but how can one show the forces are equally strong?
For this you need Newton's 2nd law, and here is how.
But first, a few words about notation. Both Newton's 2nd and 3rd laws are vector equations, involving forces and accelerations, vector quantities which have directions as well as magnitude. The same letter in ordinary font may be used for the magnitude of that vector.
Books often use bold face letters to distinguish vectors. On the blackboard we use instead a wavy line above the letter (some teachers use an underline instead; either is OK).
The teacher may start the calculation on the blackboard, but let students finish some parts (.. .. ..) if they can.
Let us call the force F, with magnitude F, and suppose we have two objects (e.g. billiard balls) labeled "A" and "B", pushing each other apart.
Suppose "A" has mass M1 and undergoes acceleration a1
F = M1 a1
While "B" acts on "A" with force –F, so.. .. ..
While "B" has mass M2 and undergoes acceleration a2
"A" acts on "B" with force F, so
–F = M2 a2
Let us first look just at magnitude--forget the vector character, forget the minus. Then.. ..
F = M1 a1
and .. .. ..
F = M2 a2
Without the minus sign the same F appears on the left! Therefore
M1 a1 = M2 a2
Divide both equations by M2 .. .. ..
[M1 / M2] a1 = a2
and then divide both by a1 .. .. ..
[M1 / M2]
= [a2 / a1]
You can see the result: when only "A" and "B" are involved, their accelerations always have the same ratio.
Say, M1 is a heavy billiard ball collide with a light one of mass M2. If the collision is a fast one, they fly off rapidly, if a gentle one, they fly off slowly, but in each case the ratio between the accelerations is the same!
(Later on, when we discuss conservation of momentum, we will see that the velocities with which the balls rebound also have the same ratio. The reason is that both accelerations last equal amounts of time--just the times the balls are in contact. (In accelerated motion, the final velocity is acceleration multiplied by times, and if the times are equal... you get the same ratio. We will come back to that.)
-- Which ball one has the greater acceleration?
The one with the smaller mass, M2. Because then
[M1 / M2]
is more than one, and therefore, so is
[a2 / a1]
a2 is greater than a1
Now let's repeat with the minus sign: we have
–F = M2 a2
Multiply both sides by (–1), which essentially switches the minus from the left side to the right ("minus times minus is plus")
– F = – M2 a2
We already have derived the magnitudes of the accelerations. The minus sign however tells that
a2 and a1 have exactly opposite directions.
It is possible to reformulate Newton's 2nd and 3rd laws without referring to "mass" and "force"--only to the one thing which is observed, namely accelerations. Supposedly, this was the approach favored by the philosopher Ernst Mach.
In this formulation, the two laws are replaced by the following:
When two compact masses act on each other, they accelerate in opposite directions, and the magnitudes of the accelerations (for those masses) always have the same ratio.
That ratio defines the ratio of the masses, through the equation derived earlier
[M1 / M2]
= [a2 / a1]
If M2 is a liter of water (1000 cubic centimeters--definition of the kilogram), then the above equation provides a way of defining the mass M1 of the other object, "through the back door."
With mass defined, we can also define the unit of force--the Newton--as that which causes a mass of 1 kilogram an acceleration of 1 meter/second2. Thus mass and force can be defined as convenient secondary quantities, but the fundamental law only involves measurable accelerations.