Rotational Motion Let’s begin with Rotational Kinematics!! Test Wednesday March 7th Let’s begin with Rotational Kinematics!!

Let’s go out in the hall for a demo!!

Rotational kinematics Imagine that you place small coins at different locations on a rotating disk: The direction of the velocity of each coin changes continually. A coin that sits closer to the edge moves faster and covers a longer distance than a coin placed closer to the center. Different parts of the disk move in different directions and at different speeds!

Rotational kinematics © 2014 Pearson Education, Inc.

Rotational Kinematics During a particular time interval, all coins at the different points on the rotating disk turn through the same angle. So we can describe the rotational position of a rigid body using an angle.

Rotational (angular) position θ The rotational position θ of a point on a rotating object is defined as an angle in the counter-clockwise direction between a reference line (usually the +x-axis) and a line drawn from the axis of rotation to that point.

Units of rotational position The unit for rotational position is the radian (rad). It is defined in terms of: The arc length (s) The radius (r) of the circle The angle in units of radians is the ratio of s and r: To convert degrees to radians: 𝑟𝑎𝑑= 𝑑𝑒𝑔𝑟𝑒𝑒𝑠° 2𝜋

How else can arc length (s) be a linear quantity? Tip 𝑠=𝑟θ How else can arc length (s) be a linear quantity?

Measure the displacement of the hula hoop travel in one rotation Measure the displacement of the hula hoop travel in one rotation. Compare this to the circumference of the hula hoop.

What is the equation for Circumference What is the equation for Circumference? What does one time around (theta) equal? Hint: think unit circle!

How can you use theta to find displacement? Form an equation

It can be Δx! When the object makes one complete revolution, the object has moved a distance equal to the circumference, and each point on the exterior has touched the ground once. When the object rotates through an angle θ, the distance that the center of mass has moved is: ∆𝑥=𝜃𝑅

Rotational (angular) velocity ω We remember that linear velocity was the rate of change of linear position So we can also say that the rotational (angular) velocity ω is the rate of change of each point's rotational position. All points on the rigid body rotate through the same angle in the same time, so each point has the same rotational velocity. © 2014 Pearson Education, Inc.

Rotational (angular) velocity ω Constant Velocity only © 2014 Pearson Education, Inc.

Tips © 2014 Pearson Education, Inc.

The difference between linear and angular velocity For a disk or sphere rolling along a surface, the motion can be considered as a combination of rotational and translational (linear) motion The center of mass moves in a translational motion and the rest of the body is rotating around the center of mass.

Rotational (angular) acceleration α We remember that the linear acceleration is the rate of change of the linear velocity So we can also say the rate of change of rotational velocity is rotational acceleration. © 2014 Pearson Education, Inc.

Rotational (angular) acceleration α

Tip We remember that when linear velocity and linear acceleration are in the same direction the object is speeding up and that if they are in opposite directions the object is slowing down So we can also say…. © 2014 Pearson Education, Inc.

Relating linear and rotational quantities © 2014 Pearson Education, Inc.

Because linear and rotational quantities are similar, the linear and rotational graphs are as well! © 2014 Pearson Education, Inc.

Linear vs Rotational Vocab and units! Tangential Angular Radial Translational m/s rad/s m/ 𝑠 2 rad/ 𝑠 2 m rad rpm This is rotational per minute. It is a velocity BUT You will have to convert to rad/s.

Summary © 2014 Pearson Education, Inc.