## Tuesday, August 1, 2017

### Physical Quantities: Equivalency and Conversion

Objects in the physical world can be described according to various physical properties, such as length, volume, mass (or weight). Since the beginning of humanity, people have come up with ways to measure things in a standard way.

#### Standards and Units

Measurements were usually created to compare to some reference object or other agreed-upon standard. For instance, if trading sea shells, the standard would be… a seashell. One sea shell equaled one seashell.

Okay… hang in there… keep going…

So, along the way, magic occurred (not really) and people began to equate the word "one" with "unit." So, if you said, "Give me eight units of seashells," since seashells were measured in… themselves… you would get eight seashells.

But suppose a guy sold sand. Selling grains of sand would be… dumb. Suppose (sticking to the beach motif) he had a coconut hollowed out. His thing was, one scoop of sand for one seashell…

So for sand, the unit would be scoops.

A buyer would say, "Give me eight scoops of sand," and would pay eight seashells for it.

It must be time for a definition!

Unita quantity chosen as a standard in terms of which other quantities may be expressed. (Oxford dictionary.

Not very helpful, but in the right direction!

Anne Marie Helmenstine, Ph.D. (https://www.thoughtco.com/definition-of-unit-in-chemistry-605934) defines it this way:

UNIT DEFINITION
A unit is any standard used for comparison in measurements.

Going back to strictly intuition, most people already know what units are. "Comparison" is a good word to hang onto, so do that.

Now think about things you already know. Gallons of gasoline. Pounds of lunch meat. Miles to the next town.

Using a standard unit like miles allows us to make reasonable comparisons. A mile is a mile is a mile. So if it is 200 miles to Townville and 400 miles to Villaton, since a mile is a mile is a mile, it must be twice as far to Villaton.

If I have 4 gallons of paint and you have 2, I have twice as much paint, since a gallon is a gallon is a gallon.

So, think about things around you. What are the standard units for them.

Distances between towns?
Shampoo?
Foundation?
Firewood?
Soda (pop)?

Chances are, for at least one of those, you thought of different standards. For example, you might find 20 oz bottles of soda as well as 2 liter bottles!

There are times when converting from one standard unit to the other is desirable.

#### Conversion and Equivalency

The process of converting from one standard unit to another boils down to finding how many of one thing is equal to one thing of the other.

Good news, #1: The math on this is easy.

Good news, #2: You can usually look it up on Google.

Bad news, #1: An explanation follows anyway.

To start with a definition of conversion…

Conversion: The process of finding out how many units from one standard are equal to how many units from another standard.

Suppose you discover that, in a fantasy novel, the people in one town sell milk by the Nallog and in another by the Ecnuo. Some guy has a barrel on which are etched lines for both Nallogs and Ecnuos, and another dude sees that the milk comes to the 2 Nallog mark and also the 256 Ecnou mark.

In the (silly) example, you can see that 2 Nallogs = 268 Ecnous. That is, the two quantities are equivalent.

Finding out how many Ecnous are in a Nallog is easy! Divide the bigger number by the smaller. Bam!

268 ÷ 2 = 128

So 1 Nallog  (the bigger unit) is equivalent to 128 Ecnou (the smaller unit).

In the (silly) example above, the conversion factor is 2.

Conversion factor: A number that, when multiplied will convert one unit to another.

Oxford Dictionary says it like this: an arithmetical multiplier for converting a quantity expressed in one set of units into an equivalent expressed in another.

To find (or derive) a conversion factor for anything, all that is needed is to know how much of something is present in the two different units. Then, to find the conversion factor, divide the larger by the smaller of the quantities. The quotient (number after you push the equal button) will be how many of the smaller thing (the bigger number) are in the bigger thing (smaller number).

No, that's not confusing, is it?

EXAMPLE

A container has 591.5 milliliters of shampoo in it. The label also says it has 10 clarkens (which are cleverly abbreviated as "Clar"). Find the conversion factor for milliliters and clarkens.

591.5 ml
_______  =   59.15 ml/Clar
10 Clar

Thus, 1 Clar is equal to 59.15 ml.

BAM! Easy math!

#### Conclusion

It is always best to just measure things in the units you need. But sometimes, that's impossible. Sometimes your most accurate measuring device provides units that you have to convert.

A solid understanding of equivalency and conversion prepares students to face the demands of science and engineering.

## Monday, July 31, 2017

### Physical Quantities: Quantities and Units

When looking at the world around, people pretty much automatically attend to physical quantities, pretty much without thinking about them. The youngest child intuitively can judge differences with some skill.

By the time the child reaches school age, the following conversation would seem drastically out of place:

Teacher: "Mary, how far can you run?"
Mary: "I can run six pounds! Very heavy!"

Almost all children would know that far and heavy measure different physical quantities.

As most readers of this would know, one of the jobs of science is to complicate things. No… not that… To clarify things by assigning specific words with specific meanings in order to make discussion precise and accurate (which are both words surrounded by confusion, ironically).

Therefore, with regard to physical quantities, a certain set of words are used to describe specific things.

Included in those sets of words are two sides to every quantity. One side is the concept of what is being described. Mary, in the dialog above, gave an answer about one quantity with words used for a different. The other side of the concept is, for every quantity, specific ways that it is measured (which are called units). Pounds do not go with a "how far" question!

First what are the things being described?

#### Types of Physical Quantities

The following list is certainly not all-inclusive. It includes a few very common and familiar quantities that are measured.

Distance: Distance is a measure of… okay, distance is so common it actually has different meanings, so in science, there are different words that are used so in order to be more specific.
Distance (case 1): The measure of how far two points are from each other, as in, "The tip of the antenna was 12 meters from the the surface of the window."
Distance (case 2): The length of a path that a moving body takes, as in, "The dog ran from tree to tree through the park until it was at the tree next to the one where it started, covering a distance of 250 yards." Compare with Displacement, next!
DisplacementThe measure in a straight line between where a moving body begins and ends, as in, "The dog ran from tree to tree through the park until it was at the tree next to the one where it started, resulting in a displacement of only 4 yards."
Length, Width, Height: Pretty much what you expect, on this! They measure of how far specific points of an object are from other points on the object.
Other things that would be included in the distance concept are circumference, radius, perimeter, range, altitude, depth… You can probably think of others. But will you?
Volume: (Not talking about sound, which borrowed from this concept for its own purposes!) This also is pretty much what you expect!

Volume is the space in three dimensions that an object or substance takes up (or holds). If a tank holds 25 gallons of gasoline, then the volume of the gasoline that filled the tank would be (duh) 25 gallons. A bottle of soda holds 20 ounces. A different bottle holds 2 liters.

Mass: Mass is a physical quantity that is the result of how many protons, electrons, and neutrons are all in a given space. It is actually not directly observable, though there are devices (scales and balances) that combine with gravity (or other acceleration) and other laws of physics so that it can be measured. Mass is a measure of the amount of matter an object contains.

If two things with the same volume have different masses, one would feel heavier than the other.

Good news: There are scales that measure mass, so you can just plop things down and get a number!

Weight: Weight is a measure of mass under a particular condition. Many people have heard things like, "Well, on the moon, I'd only weigh 12 pounds."

Weight is a basic concept that people are very familiar with. Weight is the degree of heaviness something has.

Time: This is something people very intuitively understand, but which is actually a very abstract concept. Stop reading and write down a definition of time. Not how time is measured! Try a definition of time that does not use seconds, minutes, hours, days, etc. and see what you come up with!

According to the Oxford dictionary, time is the indefinite continued progress of existence and events in the past, present, and future regarded as a whole.

Time is a very basic concept in science, and fortunately, our intuitive understanding is enough for us to use it.

#### How do you measure these quantities?

The list below will connect the quantities above with the SOME OF THE units used to measure them. Common units in science are in bold.

Distance: inches, feet, yards, miles, meters, kilometers, centimeters, lightyears

Volume: gallon, ounce, cup, teaspoon, liters, milliliters

Mass: slugs, grams, kilograms

Weight: pounds, newtons

## Tuesday, April 25, 2017

### Formula Quick Look

The following page is a list of formulas and very brief explanations.

Density is the ratio of a substance's mass to its volume and can be expressed mathematically as

D=M/V
where D is density, M is mass, and V is volume.

Example:

What is the density of an object having a mass of 8 kg and a volume of 2 cubic meters?
D = M/V
D = 8/2
D = 4 kg/m3

Temperature:
In science, we will use Celsius or Kelvin temperature scales to describe temperature.
To convert:
Celsius = Kelvin - 273.15
Kelvin + 273.15 = Celsius

Kelvin = Celsius + 273.15
Celsius + 273.15 = Kelvin

Gas Laws:

Charles's Law
The volume of a gas is directly proportional to its temperature in kelvins if the pressure and number of molecules are constant.

V1T1=V2T2

Boyle's Law
The volume of a gas is inversely proportional to its pressure if the temperature and the number of molecules are constant.

P1V1=P2T2

Combined Gas Law
Pressure is inversely proportional to volume, or higher volume equals lower pressure. Pressure is directly proportional to temperature, or higher temperature equals higher pressure.

(P1V1)/T1=(P2V2)/T2

Example:

If a sample of gas initially has a pressure of 2 atm, a volume of 3 liters, and a temperature of 300 K, what would its final volume be if the pressure changed to 1.5 atm and the temperature changed to 290 K?
(P1V1)/T1=(P2V2)/T(2 • 3)/300 = (1.5 • V)/290
6/300 = 1.5 V/290
290 • (6/300) = 1.5 V
5.8 = 1.5 V
5.8/1.5 = V
3.87 l = V

Motion:

Finding final velocity:
vf = vi + at
where vf is final velocity, vi is initial velocity, a is acceleration, and t is elapsed time.

Example:
An object is moving at a rate of 3 m/s and accelerates at a rate of 2 (m/s)/s for 5 seconds. What is its final velocity?
vf = vi + at
vf = 3 + 2 • 5
vf = 3 + 10
vf = 13 m/s

Finding average velocity:
v(ave) = (vf+vi)/2
where v(ave) is average velocity, vf is final velocity, and vi is initial velocity.

df = di + vit + 1/2at2
where df is the final, total displacement…
di is the initial displacement. (How far from whatever point of reference is the object when the thing starts accelerating?)…
vi is the initial velocity of the object at the beginning of the acceleration.
t is the elapsed time from the beginning of the acceleration until the end of the period being observed.
(vit accounts for the motion of the object based on its starting velocity. It keeps covering distance at the initial rate, and additionally, it accelerates and covers more distance.)
a is the acceleration and t is elapsed time.

Example:
An object begins 10 meters from a mark on a track with an initial velocity of 3 m/s. If it accelerates at a rate of 5 (m/s)/s for 4 seconds, how far from the mark does it end up?
df = di + vit + 1/2at2
df = 10 + 3(4) + 1/2(5)(4)^2
df = 10 + 12 + 1/2 (5) (16)
df = 10 + 12 + 40
df = 62 m

Force:

Where F is force, a is acceleration, and m is mass, then:

F = ma

Work and Energy:

Work is found by

W = Fd
where W is work, F is force applied (not net force!), and d is displacement/distance.

Kinetic energy (KE) is found with this equation:
KE = 1/2 mv2
where KE is kinetic energy, m is mass and v is velocity.

The potential gravitational energy can be found with this equation:
PE = mgh
where PE is potential gravitational energy, m is mass, g is acceleration due to gravity, and h is height.

On earth, acceleration due to gravity is 9.8 (m/s)/s

The amount of elastic potential energy is determined by how hard it is to compress or stretch something and how far it is stretched or compressed.

The equation to find this is:
PE = 1/2kd2
where PE is potential elastic energy, k is a constant specific to a particular stretchy thing (spring, rubber band, etc.) and d is the distance that it is stretched or compressed (sometimes x is used instead of d, as in the illustration)

Heat/Energy Transfer

Now to find heat, we can use the formula:
Q = mcΔT
Where Q is heat or thermal energy, m is mass, c is a number called specific heat that you either look up or calculate, and ΔT is the change in temperature.

EXAMPLES:

How much heat is absorbed by 200 grams of water that starts at 25C and ends up at 30C, given that the specific heat of water is 4.2 J/g°C?

Q = mcΔT
Q = 200•4.2•(30-25)
Q = 200•4.2•5
Q = 4200 J

What is the specific heat of a metal that has a mass of 50 grams and changes temperature from 100C to 30C and gives off 4200 J of thermal energy?

Q = mcΔT
4200 = 50•c•(100-30)
4200 = 50•70•c
4200 = 3500c
4200/3500 = c
1.2 = c

## VARIOUS EXAMPLES

Given the following information, find the work done on a 6.5 kg object after 4 seconds:

A = 16 N
B = 4 N
C = 14 N
D = 9 N

STEP 1: Find Net Force by resolving the UpDown forces, resolving the LeftRight forces, and then using the Pythagorean Theorem:

F(net)2 = UpDown2 + LeftRight2

F(net)2 = (16 - 4)2 +(14 - 9)2
F(net)2 = (12)2 + (5)2
F(net)2 = 144 + 25
F(net)2 = 169
F(net) = 13 N

STEP 2: Find Acceleration where the force is the net force on the object:

F = ma

13 = 6.5a
13/6.5 = a
2 (m/s)/s  =  a

STEP 3: Find the distance through which the force acted due to the net force.

df = 1/2at2

df = 1/2 • 2 • 42
df = 16 m

STEP 4: Find the work done by the net force through the calculated distance.

W = Fd

W = 13 • 16
W = 208 J

To find the final velocity in the above:

Do Step 1 above.

Do Step 2 above.

STEP 3:

Vf = at

Vf = 2 • 4
Vf = 8 m/s

To find final kinetic energy, first find the final velocity (above), and then:

KE = 1/2mv2
KE = 1/2•6.5•82
KE = 1/2 • 6.5 • 64
KE = 208 J

## Tuesday, April 18, 2017

### The Transfer of Thermal Energy

http://billonscience.blogspot.com/2017/04/the-transfer-of-thermal-energy.html
Thermal energy is a relatively easy concept to grasp. Objects in nature have a temperature, and that is (by definition) the average kinetic energy of the molecules of the object. The higher the temperature, the higher the average kinetic energy. Naturally, the larger the object, the more molecules, which means more kinetic energy.

So, thermal energy is a function of how many molecules are present as well as what the temperature of those molecules are. For any given substance, the total thermal energy can be easily found.

The total thermal energy would be how much energy could be given off if the object's temperature dropped to absolute zero. This theoretical idea is not practicalInstead, thermal energy is always described as the amount of energy given off or taken in as a result of some change in temperature.

The formula for finding thermal energy given a temperature change is actually very easy, EXCEPT it relies on the idea of change in temperature. It is common to use T for temperature and likewise common (though it is possibly something new to many introductory students) to use the Δ as a symbol for "change."

Thus, Δis the symbol for change in temperature.

So, if something starts off at 100 C and ends up at 80 C, what is ΔT?

ΔT = Ti - Tf
ΔT = 100 - 80
ΔT=20

Finding Δis more about getting used to the symbol than anything else; it is the difference between starting and ending (initial and final) temperatures.

Now to find heat, we can use the formula:

Q = mcΔT

Where Q is heat or thermal energy, m is mass, c is a number called specific heat that you either look up or calculate, and Δis the change in temperature.

EXAMPLES:

How much heat is absorbed by 200 grams of water that starts at 25C and ends up at 30C, given that the specific heat of water is 4.2 J/g°C?

Q = mcΔT
Q = 200•4.2•(30-25)
Q = 200•4.2•5
Q = 4200 J

What is the specific heat of a metal that has a mass of 50 grams and changes temperature from 100C to 30C and gives off 4200 J of thermal energy?

Q = mcΔT
4200 = 50•c•(100-30)
4200 = 50•70•c
4200 = 3500c
4200/3500 = c
1.2 = c

TRANSFER OF ENERGY

Thermal energy transfer is a fairly simple concept. When (two or more) things are in the same environment (which we will call being in the same system) the molecules of the things will bump into each other until all of them have the same kinetic energy.

What does that mean? Those things with higher energy will give off their energy to the things with lower energy. After a period of equalization, everything in the system will have the same average kinetic energy.

So, suppose a hot piece of metal is put into a cup of cool water… The energy from the metal will be transferred into the water. The metal will cool off. The water will warm up.

The amount of energy given off will be equal to the amount of energy absorbed. This is a major law of physics! Energy cannot be destroyed. It can change forms, or be transferred from one thing to another, but it cannot just go away.

Therefore, if hot metal is put into cool water, the quantity of the energy lost is equal to the quantity of the energy gained. Thus, Qlost = Qgained.

SO… if you know how much energy was gained, then you know how much energy was lost.

And?

Suppose you put a sample of hot metal (for which you know the starting temperature) into water. If you know the mass of the water and you know the temperature change of the water, using the known quantity for water's specific heat (4.2), you can calculate how much energy was gained. Then, if you know the energy gained by the metal, the mass of the metal, and the change of temperature of the metal, you can then calculate the specific heat.

SUMMARY THOUGHTS

Q = mcΔT
In a system, once equilibrium is reached everything in the system will have the same kinetic energy (temperature).
Energy lost from objects in a system will equal energy gained by other objects in the system until all of the objects have equal temperature.

## Sunday, April 16, 2017

### Concepts of Force, Work, and Energy

The concepts of force, work, and energy are closely related, and they are often tied to motion. To begin understanding of how they interact, it is important to start with a firm grasp of the basic concepts.

FORCE

The starting place for understanding how force, work, and energy interact is to quickly review concepts related to force.

Force is a push or a pull that acts on something in creation.

Forces are the result of four fundamental forces found in creation: gravitational force, electromagnetic force, and the strong and weak forces associated at the atomic level.

At any given time, an object in the universe is subjected to many forces. Forces on an object add up as vectors (direction and magnitude), and if the net force is NOT zero, then the object will be accelerated at a rate given by Newton's second law:

F = ma

where F is the net force on an object, m is the mass of the object, and a is the resulting acceleration.

Force is measured in Newtons (the symbol is N), which is the potential to accelerate a 1 kg object at a rate of 1 meter per second per second. that is:
1 N = 1 (kg • m/s)/s = 1 kg • m/s2
It is important to remember that an object can be moving, but that the net force on it is zero. This does not mean there are no forces at work. It just means that it is moving at a steady rate (constant velocity) because all of the forces are balanced out. For example:
Pedaling a bicycle at a steady rate requires that the force of the tires pushing against the ground is equal to the wind resistance opposing the motion of the bike.
When a box is pushed across the floor at a steady rate, the force pushing against the box is equal to the oppositional force of friction.
Consider the box example further. The force of gravity pulls the box toward the center of the earth, causing it to press against the floor. The solid nature of the floor opposes the force of gravity, keeping the box from moving downward. The "up and down" forces are in balance, and they create the resistance of friction that opposes the force exerted on the box to push it. If the pushing force is greater than the frictional force, then the box accelerates. If they pushing force is equal, the box moves at a steady rate. If the pushing force is less than the force of friction, the box slows down or does not start moving.

WORK

Extending the concept of force to motion, the idea of work emerges.

In the context of physics, work is a very specific concept. Work is the quantity that represents the application of a force through a distance. This means that work is done when a force is exerted and when motion occurs. Pushing against a wall does no work, though it requires a lot of effort (which is a loosely used term related to work, force, and motion.

Work can be calculated as the product of force and distance/displacement:

W = Fd

where W is work, F is force applied (not net force!), and d is displacement/distance.

Work is measured in Newton-meters (the symbol is Nm or N-m) meaning that a Newton was applied through a distance of one meter.

So, back to the box example above, if the force to push the box causes it to move some distance, then work is done. If the force does not cause it to move, then effort was spent, but no work was done.

Assume the force needed to balance friction and move the box at a constant velocity is 5 N and that the box is pushed 10 m. In that case how much work was done?
W = Fd
W = 5N • 10 m
W = 50 Nm
It is vital to remember that work is done not as a result of the net force, but as a result of the applied force through a distance. In the case of a box moving at a steady rate (constant velocity) the net force on the box is zero, but the applied force is used to find work done.

ENERGY

Going to the next concept, energy, is not terribly difficult. Energy can be defined as the potential to do work.

There is, thus, a direct relationship between work and energy. However, where work is often described with the unit of Newton-meters, energy is generally described with the equivalent unit, joules (the symbol is J).

A joule is the energy needed to do 1 Newton-meter of work. So:
1 J = 1 Nm
An example will help clarify the relationship between work and energy. Think about this:

If "Cash" is the potential do "pay" and energy is the potential do work, then…
Q: If you paid \$20.00 for something, how much cash did you spend?
A: You spent \$20.00
Q: If you did 20 Nm of work, how much energy did you spend?
A: You spent 20 Nm (or 20 J)
Thus, once you have calculated the amount of work done, you know how much energy was used to do it.

TYPES OF ENERGY

Energy exists in different forms. Though it is still the potential to do work, as different forms of energy are examined, the idea of moving something through a distance can get lost. That should not be a distraction, though. Just cling to the idea that energy is the potential to do work.

Kinetic Energy—the energy of motion is called kinetic energy.

The amount of energy a moving object has can be found based on the mass and velocity of the object that is moving. Kinetic energy (KE) is found with this equation:
KE = 1/2 mv2
where KE is kinetic energy, m is mass and v is velocity.

Potential Energy—energy that is stored as a result of position or shape.

Potential energy  (PE) is a little more broad than kinetic energy. Potential energy can be thought of as how much motion could occur if the stored energy was released.

For instance, a hot wheel car at the top of the track is not moving, but if it is released, because of its position, gravity will cause it to start. The car will accelerate (overcoming friction of the track and in the bearings of the wheels) and move down the track.

Another type of potential energy relates to the shape of something. When you stretch a rubber band, because of its shape, it has stored, potential energy. Likewise a compressed or stretched spring, because of its shape, also has potential energy.

Potential energy needs to be examined more specifically.

Gravitational Potential Energy—the potential to create motion based on position within gravity and on the mass of the object.

 Source: http://www.batesville.k12.in.us/
Imagine hanging a heavy weight over a pulley and attaching to a car. Letting the weight go would create tension in the rope and that tension would create a force on the car causing it to accelerate. The higher the weight, the more potential it would have. The heavier the weight, likewise, the more potential it would have.

The potential gravitational energy can be found with this equation:
PE = mgh
where PE is potential gravitational energy, m is mass, g is acceleration due to gravity, and h is height.

On earth, acceleration due to gravity is 9.8 (m/s)/s.

 Source: http://hyperphysics.phy-astr.gsu.edu/hbase/pespr.html
Elastic Potential Energy—the potential energy of an object that is stretched or compressed.

The amount of elastic potential energy is determined by how hard it is to compress or stretch something and how far it is stretched or compressed.

The equation to find this is:
PE = 1/2kd2
where PE is potential elastic energy, k is a constant specific to a particular stretchy thing (spring, rubber band, etc.) and d is the distance that it is stretched or compressed (sometimes x is used instead of d, as in the illustration)

Regarding Potential Energy and Kinetic Energy, remember that they both deal with either the potential to create motion or deal with the actual motion.

FORMS OF ENERGY

Up to this point, energy has been looked at in terms of motion and the potential to cause motion. However, energy exists on other forms. It is important to understand that energy can change forms.

Mechanical Energy—This is the form of energy that has been discussed up to this point. Mechanical energy is the energy associated with the motion and position of everyday objects.

Thermal Energy—the total potential and kinetic energy associated with the motion of all the molecules in an object.

Understanding thermal energy relies on what was learned about the kinetic theory of matter: all objects are made up of particle that are in constant motion. When temperature increases, the molecules move faster (and take up more room).

Thermal energy is the sum of all the kinetic energy of all those molecules.

The molecules of different types of material act differently in reaction to thermal energy, but working with thermal energy is fun and relatively easy. This will be addressed later in detail.

Chemical Energy—the energy stored in the chemical bonds of a substance.

If the bonds can be broken, the energy is released. Burning is the process of creating a chain reaction of bonds breaking and giving of energy—the chemical energy is converted into light and heat.

Electric Energy—the energy associated with electric charges.

Electric energy can be converted (through use of a motor) into mechanical energy. Mechanical energy can be converted (through a generator) into electric energy. Electric energy can be converted into light and heat by a light bulb.

Electromagnetic Energy—a form of energy that travels through space as a wave.

Light is electromagnetic energy.

Nuclear Energy—the energy stored in atomic nuclei.

It takes energy to slam protons and neutrons together in a nucleus. Once the nucleus is formed, that energy is stored in the nuclear bonds. Breaking those bonds (fission) releases the energy.

SUMMARY

There is a close relationship between force, work, and energy. Energy is the potential to do work, and can be seen in different forms.

_______________

Definitions and content from:
New Oxford American Dictionary
Physical Science Concepts in Action, Pearson
http://hyperphysics.phy-astr.gsu.edu/hbase/pespr.html

## Monday, March 27, 2017

### Newton's Second Law

Newton's Second Law is one of the most famous of all things in the realm of physics. It is fundamental to so many aspects of physics, architecture, and engineering that it is hard to find anything that equally ranks in prominence.

Stated with words, Newton's Second Law says that if an object accelerates (if it changes magnitude or direction of its velocity), the force required is equal to the mass of the object times the rate of acceleration.

So, to state that in less fancy terms… Say something is being pushed. If you push something, it's going to change velocity. (That could mean starting or stopping.) So, if you recall that changing velocity is the same as accelerating, then when you push something it accelerates. The harder you push, the faster it accelerates. The heavier it is, the slower it accelerates.

Yet again, logic and common since prove true! If you just stop and think about it, you already understand this.

Just for fun, let's reprise…

If something has a lot of mass, it will change velocity less under a given force than if the same force is applied to something with less mass. The stronger the force, the faster an object will accelerate.

Naturally, since this is physics, there is a formula that models the relationship between mass, force, and acceleration. Where F is force, a is acceleration, and m is mass, then:

F = ma

This leads to a couple of derived formulas, just in case finding m or a is desired.

To find F:

F = ma

To find m:

m = F/a

To find a:

a = F/m

EXAMPLE 1:

Find the force needed to to cause a 5 kg object to accelerate at a rate of 20 (m/s)/s to the left.

F = find it
m = 5 kg
a = 20 (m/s)/s Left

F = ma
F = 5•20
F = 100 N Left

EXAMPLE 2:

If a 20 N force to the Right is applied to an object and causes it to accelerate at a rate of 4 (m/s)/s, what is the mass of the object?

F = 20 N Right
m = find it
a = 4 (m/s)/s

F = ma
20 = 4•m
20/4 = m
5 kg = m

OR

m = F/a
m = 20/4
m = 5 kg

EXAMPLE 3:

If a 15 N force to the Right is applied to a 3 kg object, what will the rate of acceleration be?

F = 15 N Right
m = 3 kg
a = find it

F = ma
15 = 3•a
15/3 = a
5 (m/s)/s = a

OR

a = F/m
a = 15/3
a = 5 (m/s)/s

Summary

Using Newton's Second Law effectively allows demonstrating the relationship between mass, acceleration, and force. It follows logic and common sense completely, and the math requires only picking out the numbers, plugging them into the equation and using multiplication and division to solve for the missing variables.

## Sunday, March 26, 2017

### Units, Variables, and Formulas

When physics formulas are considered, there are many letters to think about. The formulas, themselves, are a string of variables represented by letters. Each letter represents some quantity, and the formula shows how the quantities are related to each other.

In some formulas, the displacement is considered. Displacement (and also distance) is represented by the letter d. However, many times, it is necessary to look at multiple displacements in the same situation, subscripts are used for initial and final displacements: di and df. Often, for convenience, the subscript is displayed in the same size as the variable resulting in di and df.

In other formulas, the rate that displacement changes (velocity) is considered. Velocity is represented by the letter v. However, many times, it is necessary to look at multiple velocities in the same situation, subscripts are used for initial and final displacements: vi and vf. As with displacement, for convenience, the subscript is displayed in the same size as the variable resulting in vi and vf

When formulas consider the rate at which velocity changes (acceleration), the letter a is used.

When working with acceleration, it is sometimes necessary to look at an average velocity, which is represented as v(ave) or just v(ave). Graphically, it can be written as a v with a line on top.

Elapsed time is represented by the letter t. (It is sometimes necessary to find elapsed time by taking the difference between two clock readings, which are called—and which are confusing—initial time and final time, ti and tf).

Formulas using force will represent for with an F (capital). Working with forces often requires considering multiple forces all at once. Thus, net force becomes Fnet and the component forces get subscripted with numbers such as F1 or F2. As in other cases, for typing convenience, the subscript is sometimes rendered in the same size font as the variable.

One more variable that relates to force and acceleration is mass, and it is represented by an m. If necessary to designate different masses in the same situation, subscripts are used to designate which is which.

The following formulas give examples of how the variables might appear in equations.

df = di + vit + 1/2at2

vf = vi + at

F = ma

v(ave) = (vi + vf)/2

One challenge to solving physics problems is working the given information into the equations. Paying attention to the units is the key to doing this successfully. The following guide will help identify where quantities with specific units should go.

Distances — df, di, d — units will be something that measures distance, such as meters (m), miles (mi), feet (ft), inches (in), centimeters (cm), etc.
Examples:
3 ft
2 m
34 cm

Time — t — units will be something that measures time, such as hours (hr), minutes (min), or seconds (s).
Examples:
3 s
4 min
65 hr

Velocity — vf, vi, v — units will be some distance unit divided by some time unit, as shown in the examples.
8 m/s
398 cm/s
4 mph
67 MPH
11 mi/s
39 ft/min

Acceleration — a — will be some velocity unit divided by some time unit, as shown in the examples.
Examples:
3 (m/s)/s
4 MPH/s
38 (m/s)/s

Force — F, Fnet, F1, F2 — will be in newtons, a derived unit of (kg•(m/s))/s. Force should also have a direction given.
Examples:
3 N Up
2 N Left

Mass — m — units will be a mass unit, such as grams (g or gm), kilograms (kg) or slugs (never mind!)
Examples:
3 gm
5 kg