AP Physics B

Summer Work:

Vectors

Note: this material assumes you are able to work with numbers that are written in scientific (or exponential) notation.

The work in this packet is due on Friday, August 31 2007.

VECTORS

Some quantities can be completely described by a single number. The mass of a block of wood, the density of steel, the temperature of a container of water, and the time it takes for a radioactive atom to decay are all examples of this type of quantity, called SCALAR quantities.

In contrast to SCALAR quantities are VECTOR quantities, or just VECTORS for short. VECTORS are quantities that have both a magnitude, or size, and a direction.

Wind velocity is an example of a VECTOR quantity. To completely describe this quantity, not only do we need to know the magnitude (11 miles per hour), but we also need a direction (from the east).

When a symbol for a VECTOR is written, it is either written in boldface or it is written with an arrow over it. In contrast, SCALAR quantities are written in italics. For example:

SCALARS		VECTORS
Temperature	T	Velocity
Mass	m	Force
Density	ρ	Magnetic field strength
Electric charge	q	Displacement	or
Frequency	f	Acceleration

Drawing VECTORS is easy: the VECTOR is shown as an arrow pointing in the proper direction.

First, though, you must fix your coordinate system. Usually, we show an x-axis and a y-axis in the picture off to the side. A dotted line parallel to the x-axis from the tail of the vector is used so we can specify the direction with a single number as an angle with the positive x-axis.

The units for these vectors, by the way, are meters per second for velocity, teslas for magnetic field strength, and newtons for force, the standard metric units for these vector quantities.

Note that the force vector is lined up on a coordinate system that does not go straight horizontal and vertical. Sometimes, when many vectors line up in a particular coordinate system along the x and y axes if they were only tilted, it is convenient to use "skewed" coordinate axes.

Vectors can be multiplied by numbers as well. The magnitude gets multiplied by the number, but the direction stays the same.

To talk about a vector without a picture, we can simply write or say "=12 m/s at 68°".

GRAPHICAL ADDITION

OF VECTORS

Vectors are numerical quantities, and just like other numerical quantities, we can perform mathematical operations with them. Often, we will want to know when several vectors add together. For instance, displacement is a vector of a distance plus a direction. If an object's motion can be described by several displacements, one after the other, then its total displacement from its starting point to its ending point is the sum of all the individual displacements.'

Let's consider a hiker as she makes a journey of several steps. First, she goes 2.0 km due N. We will represent this with a vector arrow. We chose an appropriate scale and draw it. Let's make due north line up with the positive y-axis:

1 cm = 1 km

-30°

Next, our hiker goes 1.5 km 30º south of east. To graphically add these two vectors, we will put the tail of the second vector at the head of the first vector.

Finally, we draw the vector that goes from the tail of the first vector to the head of the last vector.

-30°

From inspection of the picture, appears to be 1.8 km at 40º.

We have just performed the head-to-tail method of graphical addition of vectors. This also works with more than two vectors. Let's say that three forces act on a body: = 5.0 N at 270º, = 3.5 N at 80º, and = 2.5 N at 150º.

The total force, , appears to be around 1.7 N at 190°.

To practice these techniques, try going to such Web sites as:

http://www.walter-fendt.de/ph11e/resultant.htm

or

http://physics.bu.edu/~duffy/java/VectorAdd.html

Exercises – Set A

For each of the following sets of vectors, use a ruler and a protractor and a blank sheet of paper to draw each one head-to-tail, draw in the total from the tail of the first vector to the head of the last vector, measure its length and its angle, and report this as the sum of the vectors.

1) 4.2 cm at 30° and 6.1 cm at 125°

2) 3.1 cm at 75° and 5.6 cm at 110°

3) 5.5 cm at 165° and 7.1 cm at -55°

4) 11.0 cm at 15° and 16.5 cm at -35°

5) 6.2 cm at 70°, 5.9 cm at 165°, and 5.6 cm at 35°

6) 8.2 cm at -30°, 10.5 cm at 115°, and 11.3 cm at 40°

TRIGONOMETRY

Trigonometry deals with relationships between the parts of a right triangle. Consider a circle centered at the origin with a radius equal to 1. Draw a radius from the origin to a point P on the circle having coordinates (x, y).

The radius we draw makes an angle θ (theta) with the x-axis.

x is a function of θ: x = cos (θ), or x is the cosine of theta.

y is a function of θ: y = sin (θ), or y is the sine of theta.

If we make the circle have a radius of r, then x and y get multiplied by a factor of r.

x = r cos (θ)

y = r sin (θ)

Now we can deal with the properties of right triangles in general. Having the two equations above allows us to write the functions sin (θ) and cos (θ) as ratios of sides of a right triangle. Regard the following figure:

We now see that we can relate the trigonometric functions above as ratios of sides of a right triangle:

A third trigonometric ratio presents itself as well, the ratio of the two legs to each other:

A simple memory help: think of the right triangular mound built to in honor of Chief SOHCAHTOA of the Trig tribe.

Sine - Opposite - Hypotenuse; Cosine - Adjacent - Hypotenuse; Tangent - Opposite - Adjacent.

The last relationship between the parts of a right triangle is the Pythagorean Theorem:

There are four essential parts to a right triangle. We have labeled them x, y, r, and θ. Now we have four equations, each with one essential part missing. Having two parts, we can now find any third part asked for.

COMPONENTS

270°

If we have two vectors, one lined up on the x-axis and one lined up on the y-axis, they will graphically add up to a third vector that is at some angle to the x-axis.

As a matter of fact, any vector that is at an angle to the x-axis can be thought of as the sum of a vector lined up on the x-axis and a vector lined up on the y-axis. These vectors lined up on the x-axis and on the y-axis are called the components, or parts, of the original vector.

A vector along with its two components makes up a right triangle. This lets us use trigonometry to easily find the x-component and the y-component of any vector.

We treat the magnitude of the original vector as the length of the hypotenuse; this makes the x-component the leg that lines up with the x-axis, and the y-component the other leg that lines up with the y-axis.

For any general vector, with magnitude A, and direction θ measured from the positive x-axis, we can call the x-component and the y-component with sizes and respectively. The trigonometric functions that tell us what and are would be:

Consider a force vector, = 17 N at 125º.

We can now find the sizes of the two components, and .

Notice the negative sign on : This is okay. We keep this in front of the component. All it means is that this vector lines up on the negative x-axis. In other words, instead of a vector at 0º, it is a vector at 180º. The same thing would apply in the y-direction. A negative sign would mean that it is a component vector at 270º or -90º, instead of a vector at 90º.

Not only can we write in terms its magnitude and direction, we can also write it in terms of its components.

One way of writing a vector in terms of its components is as a sum, with the x-component denoted by the symbol (pronounced "i-hat") and the y-component denoted by the symbol (pronounced "j-hat"). If we did this, then the vector could be written as

= -9.8N + 14 N

Another way of writing a vector in terms of its components is to put the components in an ordered pair inside angled brackets. If we did this, then the vector could be written as

= <-9.8 N, 14 N>

Exercises - Set B

On a separate sheet of paper, resolve the following vectors into components. Be sure to show your work for each problem, and circle your final answer for each part.

1) = 2.5 m/s at 198º

2) = 52 N at -28º

3) = 423 m at 72º

4) = 5.5 m/s² at 273º

5) = 5.2×10⁵ m/s at 110º

6) = 2.6×10³ N/C at -39º

The sine and cosine functions let us tell what a vector's components are when we know the magnitude and direction, but we also need to go the other way. We also need to be able to tell a vector's magnitude and direction when we are given its components.

If the two components are the legs of a right triangle, and the magnitude of the vector those components make is the hypotenuse of that right triangle, then we can find the magnitude by using the Pythagorean theorem. We can also find the direction of this vector as the angle it makes with the x-axis by using the inverse of the tangent function.

Again, let's consider vector and all we know are the components, A_x and A_y. We can find the magnitude A and the direction θ by using the functions

or

or

There is one difficulty: the inverse tangent function may give you an angle that is 180º off from where it ought to be: for instance, consider the following two vectors.

If we try to find the angle for vector , we get the right number.

But, if we try to find the angle for vector , we get the wrong number.

As a matter of fact, the inverse tangent function only returns angles between 90º and -90º. If the angle is not in that range, we have to add 180º to it.

θ = -23º + 180º = 157º

The only we can tell if we need to add 180° is by seeing if the x-component is negative.

Exercises - Set C

On a separate sheet of paper, take the following components and find the total vector's magnitude and direction. Be sure to show your work and circle your final answer.

1) = 2.6 m/s² + 1.8 m/s²

2) = -1232 m + 823 m

3) = 1.2×10^-5 T + 6.2×10^-6 T

4) = <52 m/s, -34 m/s>

5) = <-2.3 N, -7.2 N>

6) = <3.2 x 10³ m, -6.9 x 10³ m>

ADDING VECTORS

MATHEMATICALLY

Adding vectors graphically is helpful for visualizing how vector quantities relate, but it does not get us the precision we need. If our scale is too small or we don't take the care we should in drawing the vectors, we get bad approximations.

The only way to accurately add vectors using mathematical tools is by following a three-step process. Let's say we wanted to add the following three force vectors: = 53 N at 75º, = 62 N at 150º, and = 78 N at -55º.

Step One: Split Each Vector into Components

Step Two: Add Like Components

	x-components	y-components
	13.7	51.2
	-53.7	31.0
	44.7	-63.9
total	4.8	18.3

Step Three: Find the Total Vector's Magnitude and Direction

Since the x-component is not negative, there is no need to add 180° to the final angle.

Therefore, the sum of 53 N at 75°, 62 N at 150°, and 78 N at -55° is 19.0 N at 75.3°.

Exercises – Set D

On a separate sheet of paper, use the technique above to take the following sets of vectors and find their mathematical sum. Be sure to show all your work, and circle your final answers.

1) 4.45 N at 35° and 3.20 N at 110°

2) 25.0 km at 15° and 61.5 km at 250°

3) 4.40×10^-3 m/s at 115° and 6.15×10^-3 m/s at 285°

4) 15.20 m at 50°, 21.60 m at 255°, and 17.20 m at 130°

5) 15.10 N at -35°, 21.65 N at 85°, and 17.27 N at 55°

6) 1.62×10³ m at 225°, 8.2×10² m at 195°, and 5.3×10² m at -65°

Answers

Use these answers to check your work. If you have any questions, be sure to send an e-mail to Mr. Poley at

michael.poley@hies.org

or check for hints on the Web site,

http://www.lightningphysics.com/vectors

Set A

1) 7.1 cm at 89°

2) 8.3 cm at 98°

3) 4.6 cm at 254°

4) 25.0 cm at -15°

5) 10.6 cm at 85°

6) 17.0 cm at 48°

Set B

1) <-2.4, -0.8> m/s

2) <46, -24> N

3) <131, 402> m

4) <0.3, -5.5> m/s²

5) <-1.8×10⁵, 4.9×10⁵> m/s

6) <2.0×10³, -1.6×10³> N/C

Set C

1) 3.2 m/s² at 35°

2) 1481 m at 146°

3) 1.4×10^-5 T at 27°

4) 62m/s at -33°

5) 7.6 N at 252°

6) 7.6×10³ m at -65°

Set D

1) 6.12 N at 65°

2) 51.4 km at -87°

3) 1.97×10^-3 m/s at 262°

4) 7.94 m at 150°

5) 36.3 N at 48°

6) 2.51×10³ m at 227°