CHAPTER 8 Rotational Kinematics

Go to this page on your laptop or computer: ◦ ms/Labs/ClassicCircularForceLab/index.html ms/Labs/ClassicCircularForceLab/index.html Experiment with different masses and lengths and observe the speed changes. Discovery Lab: Rotational Motion

Intro to Rotational Motion Go to this page on your laptop or computer: ◦ g-motion-2d g-motion-2d ◦ Click on Run Now (or Download if it is your computer and you want to have a copy of the app.) Click on the circular or ellipse motions ◦ Make note of the direction of the velocity and acceleration vectors

Discovery Lab: Rotational Motion Go to this page on your laptop or computer: ◦ Click on Run Now (or Download if it is your computer and you want to have a copy of the app.) On the intro tab: Slide the handle to rotate the table, you may pause and do this again at a different speed and direction ◦ Make note of the magnitude and direction of the velocity and acceleration vectors  You may move the bug to different positions on the table by dragging it ◦ Make note of the angle (degrees) as the table rotates clockwise or counterclockwise ◦ Observe the angular velocities.

Consider the motion of a rigid body about a fixed axis

Angular Position On your Ladybug app, click on the handle on the turntable with your mouse and move it to various positions on the circle, and observe the effect on angle. ◦ If one segment of the turntable rotates 50 degrees, how far would the other segments rotate? Move the bug to various positions, both along a radius and along a circle. ◦ Does the angle change when you move along a radius? 

Angular Position – what we need to know All the points in the object maintain the same relative position. When the object rotates, every point rotates through the same angle. The angle θ gives the angular position of every point in the object. When the object rotates, each point undergoes the same angular displacement Δ θ. 

Rotational Quantities When an object spins, it is said to undergo rotational motion The axis of rotation is the line about which the rotation occurs. It is difficult to describe the motion of a point moving in a circle using only linear quantities because the direction of motion in a circular path is constantly changing. For this reason, circular motion is described in terms of the angle through which the point on an object moves.

Rotational Quantities

8.1 Rotational Motion and Angular Displacement For a full revolution:

8.1 Rotational Motion and Angular Displacement The angle through which the object rotates is called the angular displacement.

Angular displacement (  ) Δθ = θ 2 – θ 1

Example problem #1

Example problem #2

Angular Displacement Example – Try this Two people ride on a carousel. One rides on a horse located 5 meters from the center. The other rides on a swan located 3 meters from the center. When the carousel goes around ¼ of a revolution, how far does each person travel? Horse: 7.85 m Swan: 4.71 m Where:

Angular velocity 

Example problem #3

Example problem #4

Angular acceleration ( α )

Example problem #5

Describing Rotational Motion Practice 406/1.7

Rotational motion of a rigid body is analogous to linear motion. – you may write this at the bottom of your notes if you need it Straight line motionRotation about a fixed axis Linear position x. Linear displacement  x Linear velocity v Linear acceleration a Angular position  Angular displacement  Angular velocity ω Angular acceleration α

Equations of motion for constant acceleration Straight line motionRotation about a fixed axis

Problem solving with rotational motion. Convert tangential velocities, speeds etc. to their angular counterparts by dividing by the radius. Solve for the angular motion by using the angular equations of motion just as we did with linear motion. Convert quantities to their tangential counterparts if needed.

Angular kinematics examples Toast falling off a table usually starts to fall when it makes an angle of 30 degrees with the horizontal, and falls with an angular speed of where l is the length of one side. On what side of the bread will the toast land if it falls from a table 0.5 m high? If it falls from a table 1.0 m high? Assume l = 0.10 m, and ignore air resistance. When a turntable rotating at 33 rev/min is turned off, it comes to rest in 26 s. Assuming constant angular acceleration, find the angular acceleration and the angular displacement. If the turntable is 0.20 m in radius, how far would an ant riding on the outside edge have moved in that time?

Rotational Kinematics Practice 406/ /1.7

In comparing angular and linear quantities, they are similar. you may write this at the bottom of your notes you will need it

Analogies Between Linear and Rotational Motion:

SECTION ASSIGNMENT DUE TOMORROW

SECTION 2: Tangential and centripetal acceleration

Tangential speed Tangent line

Angular and Tangential Quantities ◦ Each angular quantity has a corresponding tangential quantity.  The arc length s is the distance travelled along the circle, in m.  The tangential velocity is the speed of the particle in the direction tangent to the circle, in m/s.  The tangential acceleration is the acceleration of the particle tangent to the circle, in m/s 2.

Angular and Tangential velocities Since s = r  v t  r  

Example problem #1

Rotation and Centripetal Acceleration If an object is rotating about a fixed axis, even at a constant speed, every point in that object is undergoing a centripetal acceleration as well.

Tangential acceleration

Centripetal acceleration

Centripetal Acceleration in Angular Form

Total acceleration The total acceleration is the vector sum of centripetal acceleration, which points to the center of the circle, and tangential acceleration, which is tangent to the circle. atat aCaC Sketch this in your notes!!

Example problem #2

Speed and Period Since the speed we are considering is constant, we can calculate it by dividing the distance for one revolution, i.e. the circumference, by the time for one revolution, which is called the period.

Rolling Objects – center of mass The center of mass of a rolling wheel will have the same velocity and acceleration as a point on the edge of the wheel.

Rolling Objects

SECTION ASSIGNMENT