SAT II Physics (Gary Graff) Episode 1 Part 3 doc

• The region between the x and y coordinates on the upper right is called the first quadrant.. • The region between the x and y coordinates on the upper left is called the second quadran

Chapter 1

PRELIMIN

CHAPTER 1

PRELIMIN PRELIMINAR AR ARY CONCEPTS Y CONCEPTS

SIMPLE EQUA

SIMPLE EQUATIONS AND ALGEBRA TIONS AND ALGEBRA

Physics is the branch of science that studies physical phenomena.Why does the sound of a starter’s pistol reach an observer standing at

the race finish line after the puff of smoke is seen when the gun is

fired? Why do some objects float in water, while other objects sink?What causes a ball to roll downhill?

These questions and more are the stuff of physics Sometimes thestudy of physical phenomena involves observation, experimentation,and calculation The calculations used in this review are accomplished

by using a little algebra, a little geometry, and a little trigonometry.Many relationships in physics can be expressed as equations Forexample:

the equation We must perform a little algebraic manipulation to

isolate a so that it is the only quantity on its side of the equal sign.

Before starting the process, remember one simple rule: whateveroperation you perform on one side of the equal sign, you must alsoperform on the other side of the equal sign

Let’s begin by isolating a in the equation below:

V f2 = V 02 + 2as

We then subtract V o2 from both sides:

V f2 –V o2 = V o2 –V o2 + 2as

This leads to:

f

2 0 2

2

22

The independent variable is plotted along the x axis of the graph, and the dependent variable is plotted along the y axis of the graph.

The slope for the graph is calculated by dividing the ∆Y by the

A graph is a picture of the relationship between two or more

quantities A direct relationship is one in which both quantities increase (or decrease) in the same manner In an inverse relationship, one quantity increases and the other decreases A parabolic relation-

INVERSE RELATIONSHIP

GRAPH B

Graph B shows an indirect linear relationship between x and y.

PARABOLIC RELATIONSHIP

GRAPH C

Graph C shows a parabolic relationship between x and y.

The ability to read a graph is crucial They are used to display and

compare physical concepts Be sure you can read and evaluate graphs.

GRAPHS

RIGHT TRIANGLES

Any triangle with a 90° internal angle is defined as a right triangle Insuch a triangle, the remaining two internal angles of the right trianglesum to an additional 90° If you are sure you are dealing with a righttriangle, the Pythagorean theoremPythagorean theoremPythagorean theorem becomes a useful tool whentrying to find one of the sides

The right triangle above is labeled according to convention The

side opposite the 90° angle is called the hypotenuse and is labeled c.

The side beside the smaller of the remaining two angles is called the

adjacent side b, and the side across from the smaller angle is called the opposite side a.

The Pythagorean theorem is stated mathematically as

c2 = a 2 + b2 Thus, should we know the size of any two sides, we canfind the missing side

Side Side Side

These are given as:

a c adjacent

hypoteenuse

b c opposite

adjacent

a b

Triangles 1, 2, 3, and 4 above all have an internal angle of 30°.Thus the sine, cosine, and tangent values are the same in all fourtriangles

CARTESIAN COORDINATES

The construction and use of right triangles is a valuable tool insolving physics problems Unfortunately, not all measures and quanti-ties begin neatly at zero Rather than allow this to complicate mat-ters, we simplify by using the Cartesian Coordinate SystemCartesian Coordinate SystemCartesian Coordinate System inconjunction with triangles You may also recognize this as the four

quadrant x and y system.

RIGHT TRIANGLES

• The region between the x and y coordinates on the upper right is called the first quadrant All x and y quantities in the first quadrant

are positive

• The region between the x and y coordinates on the upper left is called the second quadrant All x quantities in the second quadrant are negative, and all y quantities in the second quadrant are posi-

tive

• The region between the x and y coordinates on the lower left is called the third quadrant All x and y quantities in the third quad-

rant are negative

• The region between the x and y coordinates on the lower right is called the fourth quadrant All x quantities in the fourth quadrant are positive and all y quantities in the fourth quadrant are negative.

Y coordinate

+y +y y

Situations where the Cartesian Coordinate System is used

con-ventionally follow the x, y system for labeling purposes When a right

triangle is placed within the coordinate system, its sides can berenamed as follows:

Thus, we could look at the trigonometry functions for a righttriangle that has been placed into the coordinate system as:

y r b

c

x r

n

n changes θ= a to cos → θ=

b

y x

The conventional a, b, c labeling is changed to represent the physics application, thus becoming x, y, r, where r is the hypotenuse.

The Pythagorean Theorem now looks like this:

RIGHT TRIANGLES

UNITS AND CONVERSIONS

Oftentimes students will be given a value for a physical quantity such

as 50 km/hrand asked to perform operations that require them toconvert to m/sec Finding the correct units in this situation meansthat you must be able to change kilometers to meters and to changehours to seconds Experience has shown most students to be verycapable at performing the mathematic operation The problem seems

to be one of setting up the conversions between the differing units

50 km/hr = ? m/sec

One factor that can cause difficulty with conversions is thatmany students lack some basic information You should know in-stantly that there are 60 seconds in one minute and 3600 seconds inone hour The metric units you should know are:

Returning to the conversion mentioned, we begin:

There are 3600 seconds per hour, thus we multiply one hour by

3600 sec/hr, yielding 3600 seconds

Divide distance by time for the final value

The whole conversion can be made into a one-step calculation bywriting each part and solving.

quan-Quantities defined strictly by their magnitude are called scalarquantities Scalars are easy to recognize: a dozen eggs, a gross ofpaper clips, half a dozen apples, a kilogram of cheese All the statedquantities denote a number (12, 144, 6, 1) of the item listed, nothingmore

Therefore, a Sunday drive could easily be described by thenumber of miles traveled: “We drove 80 km today.” Should we wish to

add more information, such as where we traveled, then another

physical quantity is used

VECTORS

Quantities defined both by their magnitude and direction are called

vector quantities Vectors are also easily recognized, as they must include both magnitude and direction: four steps to the right, 50

meters south, 9N @ 45º Taking another look at the Sunday driveabove, we can define it as a vector by saying “We drove 80 km to themuseum and back today.” Vectors are always stated in respect to areference point

SCALARS AND VECTORS

SOLVING SCALAR AND VECTOR PROBLEMS

Numerical operations with scalars are simply a matter of adding orsubtracting the numbers

Example

We have 9 marbles and find 5 marbles How many do we now have?

Solution

9514

marbles marbles marbles( )+

As you can see, simply add to perform the operation

The same operation with vector quantities requires a bit morethought, as both the magnitude and direction must be included

We’ll start with a straightforward addition problem

Example

A student walks 4 blocks east and stops at an ice cream truck Afterpurchasing a snow cone, he walks another 10 blocks east Where isthe student in respect to his starting point?

Solution

4 blocks @ east(+)10 blocks @ east

14 blocks @ eastThe student has walked a total of 14 blocks due east

As in this example, when two vectors are added together, the

result is conveniently called the resultant vector The original vectors, which were added together, are the component vectors.

The problem above shows the nature of vectors Both magnitude

and direction must be included in operations with vectors

The next problem requires the use of the Pythagorean Theoremwhen we add a pair of vectors We will also use the Cartesian Coordi-

nate System (x and y axis).

Example

A bird is perched in the tree where it has its nest The bird flies 500 mdue east and lands on the ground in a field where it finds a worm.When the bird takes off, it is chased by a hawk, so the bird flies 300 mdue north before landing in a tree

What direction must the bird fly to find its nest, and how faraway is the nest?

Solution

When solving physics problems, always use a convenient method withwhich you are comfortable This is how you should approach anyproblem you are solving A method is suggested below

1 Draw a diagram (above)

2 Isolate and label the parts of the problem

3 Identify both x and y components

4 Write the equation

5 Solve for the unknown

Let’s place the bird’s starting point at the x, y juncture, which is

also its starting point at the nest The bird is presently located at thehead of the 300 m north vector

Before solving the problem, convert the compass values to the

SCALARS AND VECTORS

As we inspect the problem it becomes clear that the PythagoreanTheorem is the best way to solve the problem.

c2 = a2 + b2 Change to the x, y coordinates and

c2 = y2 + x2 identify c (the hypotenuse) as the resultant

Thus the resultant vector (where the bird is located in respect toits starting point) is

583m @ 31°

Recall that the bird wanted to fly back to its nest It can’t fly at31° from its current position because that path takes the bird fartherfrom its nest The direction we have found must be reversed for thebird to return to its nest

The resulting direction is exactly 180° opposite the direction thebird must fly Take the resultant vector and add (or subtract) 180°to

or from the vector’s direction

180° + 31° = 211°

The bird must fly 583 m @ 211° from its present position toreach its nest When this is done, the bird will effectively cancel outthe resultant vector That’s why its flight will be the equilibrant

An equilibrant vector is a vector that is exactly equal in tude and opposite in direction from the resultant vector.

magni-Example

Let us look at one more vector problem

Suppose 4 ropes were used to pull on a stationary object

• Rope a pulls in a direction of 15° with a force of 25N.

• Rope b pulls in a direction of 215° with a force of 16N.

• Rope c pulls in a direction of 75° with a force of 20N.

• Rope d pulls in a direction of 300° with a force of 30N.

If a single rope were to replace the four ropes, with what forceand in what direction must the rope pull?

Solution

The diagram above allows us to see each vector in relation to all theother vectors

We will isolate each vector in its turn and break each one into its

x and y components Then we can combine all the individual x and y

components to find the resultant vector

SCALARS AND VECTORS

Rope a extends 15° into the first quadrant.

Side y = Side r (sin 15°) = (30N) (.26) = 7.8N

Side x = Side r (cos 15°) = (30N) (.97) = 29.1N

Both components of rope a are located in the first quadrant.

Rope b extends 35° into the third quadrant

Side y = (side r) (sin 35°) = (16N) (.57) = 9.1N Side x = (side r) (cos 35°) = (16N) (.82) = 13.1N Both components of rope b are located in the third quadrant.

y x

negative N negative N

SCALARS AND VECTORS

Rope c extends 75° into the first quadrant.

Side y = (side r) (sin 75° ) = (20N) (.97) = 19.4N

Side x = (side r) (cos 75° ) = (20N) (.26) = 5.2N

Both components of rope c are located in the first quadrant.

Rope d extends 30° into the third quadrant.

Side y = (side r) (sin 60°) = (30N) (.87) = 26.1 Side x = (side r) (cos 60°) = (30N) (.5) = 15.0N Both components of rope d are located in the fourth quadrant.

y x

negative Npositive N

26 1

15 0

SCALARS AND VECTORS

All four ropes have been broken into their component vectors

at this point It is time to add them up We can accomplish this by

using a simple x, y chart The chart is filled in with the individual

components we have just found Having done that, we then

algebra-ically combine all the y components and do the same with the

X axis

+29.1N 13.1N+ 5.2N+15.0N

The resultant vector is 37N @ 359°

Simply stated: One rope pulling with a force of 37N in a

CHAPTER SUMMARY

• Scalar quantities are defined by their magnitude only They aretreated simply as numbers in the mathematics operationsinvolving them

• Vector quantities are defined by both their magnitude anddirection Mathematics operations involving vectors requireboth the magnitude and the direction to be included

• The direction of a vector may be implied, given as compassheadings, or given as Cartesian Coordinates

• A resultant vector is the vector formed when two or moreother vectors are combined

• Every vector may be broken into its x and y components.

• An equilibrant vector is a vector whose magnitude is exactlyequal to and opposite in direction of a given vector

CHAPTER SUMMARY

Chapter 2

MECHANICS