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Directions – Pointed ThingsPowerPoint Presentation

Directions – Pointed Things

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**luz** - Follow User

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Directions – Pointed Things. January 23, 2005. 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.

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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 *&@$ am I??

A

Vectors

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 with Unit Vectors

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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