Lesson Plan #27         http://www.phy6.org/stargaze/Lnewt2nd.htm

# (18)  Newton's Second Law

A short section, recapitulating earlier ideas of force and inertia (also known as mass). It summarizes Newton's laws, defines the unit of force known as "Newton," and gives as a practical application of the second law the acceleration of the German V-2 rocket, used in World War II.
(Revised lesson plan, following a revision of section #18).

Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern

 This lesson plan supplements: "Newon's Second Law," section #18           http://www.phy6.org/stargaze/Snewt2nd.htm "From Stargazers to Starships" home page: ....stargaze/Sintro.htm Lesson plan home page and index:             ....stargaze/Lintro.htm

Goals: The student will

• Review Newton's laws of motion.

• learn that physical quantities and laws depend on units used for 3 basic measurable quantities--distance, mass and time. Physicists nowadays base their calculation on the meter, kilogram and second, resulting in the so-called MKS system. As long as you stick with the system, all your equations and results are consistent.
• Learn that Newton's 2nd law F=ma (bold-faced symbols are now vectors) may sometimes be more appropriately viewed as a=F/m.
• Go through a sample calculation on the acceleration of the V-2 rocket used in World War II.

Terms: Mass, including inertial and gravitational mass, force, newton (unit of force), MKS system.

Review:

--What do you understand by force?
--Give an example of a force

Forces can be opposed or unopposed.

-- What happens when a force acting on an object is opposed?

It may be balanced by an equal force, and then no motion occurs. Example: Book on top of a table, its weight stopped by the equal opposing resistance of the table.
Or it may overcome the opposing force, and then work is performed, and energy is invested.

-- What happens when a force acting on an object is not opposed to any significant degree?

The object accelerates.

--What determines how big that acceleration will be?
--The mass of the object.
[Any other word for "mass"? Yes, "inertia"]
--The weight of the object. [But weight is the result of gravity. Does gravity control the acceleration?
No, it does not, the "mass" or "inertia" does that. However, weight is also proportional to the mass, so "weighing" an object is one way of determining its mass. It does so, however, by using gravity.
--Can mass be determined without using gravity?
[Yes--the inertial balance used on "Skylab" did so, as does the experiment with a hacksaw blade.]

A big stone is many times heavier than a small pebble. It is pulled down by a much stronger force.

Yet stone and pebble fall at the same rate! Why doesn't the stone fall faster?

--Because the inertia resisting its fall is larger. If it is 100 times heavier than the pebble, its inertia or "mass" is 100 times larger too, and this offsets the larger force.

### The MKS System

-- All physical measurements depend on the units we use for 3 fundamental quantities. What are they?

--Length
--Mass
--Time
The MKS system is a formulation of the laws of physics, in which--
• all distances are measured in meters (M)
• All masses are measured in kilograms (K)
• Ass times are measured in seconds (S)
As long as we stay within the MKS system, the units fit each other. For instance: the energy unit in MKS is the joule, so as long as we stay with MKS, all energies always come out in joules, no matter what area of physics we work with. (Note: Calories are independent units, not part of the MKS system).

Example:

• The unit of distance is the meter
• The unit of velocity is therefore the meter per second.
How much is 10 miles per hour? A mile equals 1609 meters, 10 miles are 16,090 meters, an hour contains 3600 seconds, so if you move at 10 mph you cover

16,090 meters in 3600 seconds
16,090/3600 = 4.469 meters in 1 second
10 mph = 4.469 meters/sec
1 mph = 0.469 meters/sec
600 mph = 268.17 meters/sec

An object undergoes "one unit of acceleration" if its velocity increases 1 meter/second each second. We therefore refer to it as "1 meter/second2" or in short "1 m/s2".

A freely falling stone accelerates at about 9.81 m/s2

The MKS unit of force is called the Newton.

It is the force which gives 1 m/s2 to 1 kg of mass.
The weight of a stone of mass 1 kg is 9.81 Newton. It is a force which, acting freely, gives it an acceleration of 9.81m/s2

--In the rest of this lesson we assume as an approximation g = 10 meter/sec2. If your body weighs 70 kilograms--and presumably, also has 70 kilogram of mass--what is your weight in newtons?

When you jump from a high place, gravity gives your body an acceleration of g=10 m/s2. Your weight is therefore 700 newtons.

If bold face letters denote vectore, we can write Newton's 2nd law as
F = ma
or as
a = F/m
Which one do you prefer?

Both of course say exactly the same thing, each can be derived from the other. But if force is the cause and acceleration the resulting effect, the second form makes more sense--given the force, given the mass, we want to find the acceleration.

Note: the example below is also on the web page Snewt2nd.htm. It was placed there when the page was revised.

-- The V2 rocket in World War II had a thrust of about 240,000 newtons and a mass of 12 tons or 12,000 kilograms. What was its upward acceleration at launch? (Solve on the board, though a student may do the writing and participate in the solution.)

This is a tricky question, related not only to F=ma but also to the concept of equilibrium, discussed at the end. An unthinking application of Newton's second law would give

a = F/m = 240,000/12,000 = 20 m/s2 = 2g

but is wrong.

Before launch, the rocket's weight is supported by the launching pad. Its weight is 12,000 g = 120,000 newton and since it does not move, an equal and opposite upward force of 120,000 newtons is exerted on it by the pad from below.

At the lift-off moment, that force ceases to act on the rocket: instead, the thrust of the engine now supports the rocket's weight (and if the engine generates a thrust smaller than the weight--less than 120,000 newton--the rocket will not lift off). So that force must be subtracted from what is available to accelerate the rocket. The result is

a = F/m = (240,000 - 120,000)/12000 = 10 m/s2 = 1 g

--At burn-out, the V2 has consumed 9 tons of fuel. What is its final acceleration just before that moment?

The total mass left is 3000 kg, the total weight is 30,000 newtons, so a = F/m = (240,000 - 30,000)/3000 = 210,000/3000 = 70 m/s2 = 7g.

--In some weird alternate universe, weight and mass are not proportional. Two materials, astrite and barite, have the same weight per unit volume, but a volume of astrite has twice the mass of a similar volume of barite. Assuming the inhabitants play a game similar to bowling--which of the two would be a better material for bowling balls? (have a discussion).

Probably astrite. Balls of equal size and shape of the two materials are equally hard to lift, but astrite needs a greater force to get started and therefore delivers a greater force to the pins.

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Author and Curator:   Dr. David P. Stern
Mail to Dr.Stern:   stargaze("at" symbol)phy6.org .

Last updated: 10-15-2004