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## PowerPoint Slideshow about 'Directions – Pointed Things' - luz

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### The following slides were supplied by the publisher ……………… thanks guys!

Today …

- We complete the topic of kinematics in one dimension with one last simple (??) problem.
- We start the topic of vectors.
- You already read all about them … right???
- There is a WebAssign problem set assigned.
- The exam date & content may be changed.
- Decision on Wednesday
- Study for it anyway. “may” and “will” have different meanings!

Vectors

One more Kinematics Problem

A freely falling object requires 1.50 s to travel the last 30.0 m before it hits the ground. From what height above the ground did it fall?

h=??

30 meters

1.5 seconds

Vectors

The Consequences of Vectors

- You need to recall your trigonometry if you have forgotten it.
- We will be using sine, cosine, tangent, etc. thingys.
- You therefore will need calculators for all quizzes from this point on.
- A cheap math calculator is sufficient,
- Mine cost $11.00 and works just fine for most of my calculator needs.

Vectors

Hey … I’m lost. How do I get to lake Eola

NO PROBLEM!

Just drive 1000 meters and you will be there.

B

HEY!

Where the *[email protected]$ am I??

A

Vectors

YUM!!

Vectors

Vectors

Vectors and Scalars

- A scalar quantity is completely specified by a single value with an appropriate unit and has no direction.
- A vector quantity is completely described by a number and appropriate units plus a direction.

Vectors

Vector Notation

- When handwritten, use an arrow:
- When printed, will be in bold print: A
- When dealing with just the magnitude of a vector in print, an italic letter will be used: A or |A|
- The magnitude of the vector has physical units
- The magnitude of a vector is always a positive number

Vectors

Vector Example

- A particle travels from A to B along the path shown by the dotted red line
- This is the distance traveled and is a scalar
- The displacement is the solid line from A to B
- The displacement is independent of the path taken between the two points
- Displacement is a vector

Vectors

Equality of Two Vectors

- Two vectors are equal if they have the same magnitude and the same direction
- A = B if A = B and they point along parallel lines
- All of the vectors shown are equal

Vectors

Coordinate Systems

- Used to describe the position of a point in space
- Coordinate system consists of
- a fixed reference point called the origin
- specific axes with scales and labels
- instructions on how to label a point relative to the origin and the axes

Vectors

Cartesian Coordinate System

- Also called rectangular coordinate system
- x- and y- axes intersect at the origin
- Points are labeled (x,y)

Vectors

Polar Coordinate System

- Origin and reference line are noted
- Point is distance r from the origin in the direction of angle , ccw from reference line
- Points are labeled (r,)

Vectors

Polar to Cartesian Coordinates

- Based on forming a right triangle from r and q
- x = r cos q
- y = r sin q

Vectors

Cartesian to Polar Coordinates

- r is the hypotenuse and q an angle
- q must be ccw from positive x axis for these equations to be valid

Vectors

Example

- The Cartesian coordinates of a point in the xy plane are (x,y) = (-3.50, -2.50) m, as shown in the figure. Find the polar coordinates of this point.
- Solution: From Equation 3.4,

and from Equation 3.3,

Vectors

If the rectangular coordinates of a point are given by (2, y) and its polar coordinates are (r, 30), determine y and r.

Vectors

Adding Vectors Graphically

Vectors

Adding Vectors Graphically

Vectors

Adding Vectors, Rules

- When two vectors are added, the sum is independent of the order of the addition.
- This is the commutative law of addition
- A + B = B + A

Vectors

Adding Vectors

- When adding three or more vectors, their sum is independent of the way in which the individual vectors are grouped
- This is called the Associative Property of Addition
- (A + B) + C = A + (B + C)

Vectors

Vector A has a magnitude of 8.00 units and makes an angle of 45.0° with the positive x axis. Vector B also has a magnitude of 8.00 units and is directed along the negative x axis. Using graphical methods, find (a) the vector sum A + B and (b) the vector difference AB.

Vectors

Subtracting Vectors

- Special case of vector addition
- If A – B, then use A+(-B)
- Continue with standard vector addition procedure

Vectors

Multiplying or Dividing a Vector by a Scalar

- The result of the multiplication or division is a vector
- The magnitude of the vector is multiplied or divided by the scalar
- If the scalar is positive, the direction of the result is the same as of the original vector
- If the scalar is negative, the direction of the result is opposite that of the original vector

Vectors

Each of the displacement vectors A and B shown in Fig. P3.15 has a magnitude of 3.00 m. Find graphically (a) A + B, (b)

A B, (c) B A, (d) A 2B. Report all angles counterclockwise from the positive x axis.

Vectors

Components of a Vector

- A component is a part
- It is useful to use rectangular components
- These are the projections of the vector along the x- and y-axes

Vectors

Unit Vectors

- A unit vector is a dimensionless vector with a magnitude of exactly 1.
- Unit vectors are used to specify a direction and have no other physical significance

Vectors

Unit Vectors, cont.

- The symbols

represent unit vectors

- They form a set of mutually perpendicular vectors

Vectors

Unit Vectors in Vector Notation

- Ax is the same as AxandAy is the same as Ayetc.
- The complete vector can be expressed as

Vectors

Adding Vectors Using Unit Vectors – Three Directions

- Using R = A + B
- Rx = Ax + Bx , Ry = Ay+ By and Rz = Az + Bz

etc.

Vectors

Consider the two vectors and . Calculate (a) A + B, (b) AB, (c) |A + B|, (d) |AB|, and (e) the directions of A + B and AB .

Vectors

Three displacement vectors of a croquet ball are shown in the Figure, where |A| = 20.0 units, |B| = 40.0 units, and |C| = 30.0 units. Find (a) the resultant in unit-vector notation and (b) the magnitude and direction of the resultant displacement.

Vectors

The helicopter view in Fig. P3.35 shows two people pulling on a stubborn mule. Find (a) the single force that is equivalent to the two forces shown, and (b) the force that a third person would have to exert on the mule to make the resultant force equal to zero. The forces are measured in units of newtons (abbreviated N).

Vectors

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