Rotation and Revolution In this unit we will be investigating objects moving in a circular path about an axis. We will see two types of motion: – Rotation An object spinning on its own axis. The axis of rotation is inside the object. – Revolution An object in motion around an axis. The axis of rotation is outside the object.

Frequency and Period Frequency – The number of rotations or revolutions every second. – SI Unit is Hertz (s -1 ) but can also be measured in rotations per second, revolutions per minute (rpm), etc. Period – The time it takes to complete one rotation or revolution. – Measured in Seconds Frequency and Period are inverses of one another.

What is a radian??? We will watch this video twice. The 2 nd time write a short summary of what this video means to you.

Think Pair Share Turn to your elbow partner The person wearing the most buttons: – Take two minutes to share your written main idea The 2nd partner: – Take two minutes to share your main idea Each of you add to your summary as appropriate

Another way to think of  … What is the equation for circumference of a circle? How do you get half of the circumference of a circle? How do you get 1/3? Do you see a pattern? The arc length of a circle is always

We use “x” to show displacement in meters – Use θ to show displacement in radians We use “v” for velocity in m/s – We will use ω for velocity in radians/s Linear to Angular Equivalents

LinearAngular Displacement x  Velocity vω Linear to Angular Equivalents

Angular Velocity A measure of the rate at which an object is rotating. How many radians an object is rotated in a certain amount of time. Equations – ω (“omega”): Angular Velocity – ∆θ : Angular Displacement (in radians) – t: time (in seconds) – T : Period (in seconds) SI Units are radians/second

Angular Velocity FYI Sometimes you will hear angular velocity referred to as: Angular Frequency They are the same thing!!

Example 1 A merry-go-round is in constant motion and a 35 kg child sitting on the edge completes one revolution every 25 seconds. a)What is the period of rotation of the merry-go-round? b)What is the frequency of rotation of the merry-go-round? c)What is the angular velocity of the merry-go-round? d)What is the angular displacement of the merry-go-round after it has been rotating for 5 minutes?

Angular and Tangential Velocity Question: A spinning carousel has a black horse that is located 1 meter from the center and a white horse that is 3 meters from the center. – Which horse has greater angular velocity? Neither! They have the same angular velocity because they are both rotating with the carousel. Angular velocity is not dependent on the radius of the circle. – On which horse would a rider “feel” like they are going faster? The white horse. It covers more circumference in the same amount of time as the black horse. It has greater tangential (linear) velocity.

Tangential (Linear) Velocity The speed of an object at a certain position from the center of the circular path. Directed “tangential” to the circular path it travels. Dependent on angular velocity and the radius of the circle. Equation – v T : tangential velocity (in m/s) – r : radius of the circular path (in meters) – ω : angular velocity (in rad/s) Uniform Circular Motion occurs when objects move in a circular path at a constant speed.

Example 2 A large circular platform is rotating at a constant speed and has two people standing on it. Jaylyn is 2 meters from the center while Chris is 5 meters from the center. The platform completes one rotation every 9 seconds. a)What is the angular velocity of the platform? b)What is Jaylyn’s tangential velocity? c)How fast is Chris moving?

Centripetal Acceleration Question: An object is rotating in a circle with a constant linear speed. Does it have a constant velocity? – No! The direction of the velocity is constantly changing. The object must be accelerating. This acceleration is directed toward the center of the circle, and is called Centripetal Acceleration. “Centripetal” means “center pointing” Equation – a c : centripetal acceleration (in m/s 2 ) – v T : tangential velocity (in m/s) – r : radius of the circular path (in meters)

Example 3 An 800 kg car drives at a constant speed of 25 m/s around a circular stretch of road that has a radius of 30 meters. a)What is the centripetal acceleration of the car? b)What is the angular velocity of the car?

Centripetal Force In order for an object to move in a circular path, a centripetal force is required. A centripetal force could be any kind of force (applied, normal, tension, gravity, etc.) Equation – Derived from Newton’s 2 nd Law – F c : centripetal force (in Newtons) – m : mass (in kg) – v T : tangential velocity (in m/s) – r : radius of the circular path (in meters) Without a centripetal force, the object would fly off in the direction of the tangential velocity.

Example 4 A 1,100 kg truck drives at a constant speed around a circular stretch of road that has a radius of 35 meters. It is found that the force of static friction keeping the tires from sliding toward the outside of the curve is 9,500 Newtons. What is the speed of the truck?

Example 5 A student swings a 3.5 kg rock in a horizontal circle at the end of a string that is 1.24 meters long. How much tension is required in order for the student to swing the rock so that it completes 2 revolutions per second?

“Centrifugal” Force The “force” that “pulls” toward the outside of the circular path. Not a real force! A force is an interaction between two objects. It is really caused by Inertia (tendency to remain in motion) As something is turning, inertia causes it to want to continue forward, and that results in the feeling of a “force”.