Distance and Displacement

Distance: How far an object travels to get from one point to another. This has just a magnitude.

It is a scalar.Displacement: The overall (RESULTING) change in position only. Displacement = PositionFinish – PositionStart This has BOTH magnitude and direction It is a Vector.

VECTORSHave both a magnitude and a directionUsed to represent velocities, acceleration, forces, anything that has both a magnitude and a direction.Vectors can be added, but their directions must be considered.The sum of two or more vectors is called a RESULTANT.A resultant is the combination of 2 or more vectors.When numbers are added you get a result, when vectors are added you get a resultant

First Example Distance and Displacement

Tommy walks around a square table 2 meters in length on one side. How far did Tommy walk? (How far asks for a distance)

Start/Finish Walked 2 m

Walked 2 m

Walked 2 m

Walked 2 m

The distance Tommy walked is 2m+2m+2m+2m

or simply8 meters

Tommy’s Displacement is 0 meters. That is because Tommy finished where he started. In the end there was one changein his position.

What is Tommy’s Displacement?

Second Example of Distance and Displacement

A girl standing still throws a ball upwards 6 meters in the air, and catches it on it’s return at the same height it was thrown.How far did the ball travel? (Looking for a distance again)

6 metersup

6 metersdown

The ball travels a total distance of 12 meters.

The displacement here again is zero meters. The ball landsat the same spot it was thrown. In the end the ball’s position did not change.

What is the ball’s Displacement?

Last exampleof Distance and Displacement

Jean runs 10 meters to the East, stops and then runs4 meters to the west. What is Jeans displacement?

Runs 10 meter East

Runs 4

metersWest

Start

Finish

Displacement(A.K.A.)

Resultant

The displacement is6 meters to the East

The distance traveled is simple 14 meters.

Rotational vs. Circular motion

When a planet orbits a star, or when a car makes a turn we have objects that can be thought of a single point that is moving around another point that is outside the object.

Radius ofmotion

Velocity of object

Path of motion

Center of motion(outside the moving object)

When a planet rotates on it axis, or when a wheel spins around its own center, we can view each of the nearlyinfinite number of points in these objects going through circular motion (not a fun idea) or instead we can view them as a single object going through rotational motion (an object spinning around a point inside the object itself)

Center of motion(inside the moving object)

Rotation ofobject

The need for a different prospective on displacement

If we have a disk that rotates one time, what is the displacement of a point on the disk, and what is the displacement of the disk itself?

Based on our Cartesian system thedisplacements of both are that there is no displacementSo if we want to describe the rotation ofthe disk (or any point on the disk) we need a system that measures angles.

Another position SystemIf we look at the disk as a whole and the angle that the disk rotates through we will have a non-zero value, even though the individual points on the disk return to there starting point.

= 90O = 180O = 270O = 360O

So for rotational motion we describe a disk’s position using an angular system.We will call the disk’s orientation (the position of a point on the disk) its angular position ().Important Note: There is no limit to how big can be. For trigonometry the sine, cosine, and tangent function values repeat in cycles from 0O to 360O. Sois some cases 0O can be considered equivalent to 360O, but that does not makethem equal.

Angular displacement ()

So even though we are using a new system to measure (or describe) an object’s position none of our definitions and relationships such as displacement, velocity, speed, acceleration , momentum, work, or energy change

To make a distinction from the displacement we have always worked with in our “traditional” Cartesian system (Pos) and this angular displacement we always use The complete term “Angular Displacement”.

Angular Displacement = change in Angular Position

By definition displacement is the change in position, and definitions always apply (otherwise they would not be a definition)

Angular displacement part 2

Reference line to measure positions from.Angular Position: = 30O clockwise

Reference line to measure positions from.Angular Position: = 30O Counterclockwise

The definition of displacement does not changeSo we need to give the magnitude of the angle, but also state a direction such as clockwise or counter clockwise

Units for Angular displacement

Any of the following can be used to describe angular displacement

1) Revolutions (1 complete circle)2) Degrees (360 degrees = 1 Revolution)3) Radians (2 Radians = 360 degrees = 1 Revolution)4) Gradians (a British unit for angles

there are 400 gradians in 1 circle (or revolution)