Two Dimensional MotionCircles and Projectiles

Two Dimensional Motion• 1 - Motion in Two Dimensions

• 2 - Circular Motion

• 3 - Centripetal Force, Gravity, and Orbits

• 4 - Center of Mass

1 - Motion in Two Dimensions • Review of Vectors

• Vectors have magnitude (length) and direction

• Without direction, the magnitude is called a scalar value

• Speed vs. Velocity

Projectile Motion• Motion is broken into two

separate velocity vectors

• X

• Y

• The combination of X and Y gives you the object’s path, or TRAJECTORY

Human Accuracy of a Projectile• What factors determine where the

ball goes?

• Angle of release

• Speed

• Determine TRAJECTORY

Human Accuracy of a Projectile

Watch how accurate Drew Brees is.

Human Accuracy of a Projectile• What factors determine where the

ball goes?

• Angle of release

• Speed

• Determine TRAJECTORY

Ball Rolling off a Table

Ball Rolling off a Table – X axis

Ball Rolling off a Table – Y axis

Distances Travelled

Range of a Projectile

45° will give you the best distance.

2 - Circular Motion

Definitions• Axis – the line that an object turns around• Rotation – when the axis is inside the object• Revolution – when the axis is external

Rotates

Revolves

Angular Speed

• Angular Speed is the number of spins in an amount of time

• Usually revolutions per second

Angular Speed, Linear Speed, and Distance• Two people sitting at different places on the same

merry-go-round have the same angular speed, but the person sitting farther from the center has the faster linear speed

Linear Speed Example

• The blades on a ceiling fan spin at 360 RPM. The fan has a radius of 0.5 m. Calculate the linear speed of the tip of the blades in m/s.

• Looking for: linear speed in m/s• Given: angular speed in rpm and radius in meters• Relationship: • Solution: The blades spin at 360 rpm, so it

takes .16 seconds to make 1 rotation

Period and Frequency• Period, T – The time it takes to complete one cycle• Frequency, f – How many cycles are completed in

one second

T = f = Example:If a fan spins around 5 times in one second, each spin takes .2 seconds

f = 5 cycles per second (Hertz) T = .2 seconds per rotation

3 Centripetal and Centrifugal Forces

Centripetal Force

• Any force that causes an object to move in a circle

• Always perpendicular to the direction of motion

• Always toward the center of the circle

Centripetal Force, Inertia, and Velocity

The object wants to continue in a straight line because of its inertia.

Its only because of whatever is creating the centripetal force (a string, rubber band, etc.) that the object moves in a circle.

Lateral Acceleration• Acceleration towards the center of a circle:

For comparison:• In a straight line, a = 0• Gravity, g = 9.8 m/s2

Calculating Centripetal Force• Combine the acceleration formula with Newton’s

Second Law

F = ma

F = m()

Example

A 4.0 kg object is traveling in uniform circular motion of radius 2.0m. The magnitude of its velocity, its speed, is 15 m/s. Determine its acceleration. Determine the net force acting on it.

•

• a = • a = • a = 112.5 m/s2 towards the center of the circle

Example Part 2

Determine the net force acting on it.

• Fnet = ma

• Fnet = (4kg)(112.5 m/s2)

• Fnet = 450N

Example of Velocity

An object is traveling in a circle of radius 4m and completes five cycles in 2s. What are its period, frequency and velocity?

• T = • T = • T = 0.4s

• v = • v = • v = 62.8 m/s

Centrifugal Force• Not a real force

• Caused by inertia

• The force you feel when your body is moved in a circle

4 Center of Mass• The point around which an object

naturally spins

Mass and Center of Gravity• For an object to stay upright, its

center of mass must be above its area of support.

Time for some Top Gear…• Reliant Robin three wheeled car.

• Because it has three wheels and weighs less than 450 kg, it is taxed and licensed as a motorcycle.

• Most three wheeled cars have the single wheel in the back. This one has it in the front.