1. Rotational Motion By: Heather Britton

2. Rotational Motion Purely rotational motion - all points of a body move in a circle Axis of rotation - the point on a body that all points circle around Rotational motion requires using angular quantities Purely rotational motion - all points of a body move in a circle Axis of rotation - the point on a body that all points circle around Rotational motion requires using angular quantities

3. Rotational Motion To know how far a body has rotated we measure the angle For rotational motion we use radians when expressing angles 2π rad = 360 o = 1 revolution To know how far a body has rotated we measure the angle For rotational motion we use radians when expressing angles 2π rad = 360 o = 1 revolution

4. Rotational Motion Angular velocity is a measure of how far an object rotates in a given time period Since all points are rotating at the same rate, the angular velocity is uniform throughout the object Angular velocity is a measure of how far an object rotates in a given time period Since all points are rotating at the same rate, the angular velocity is uniform throughout the object

5. Rotational Motion ω = Δθ / Δt ω - angular velocity (the Greek letter omega) Δθ - the change in angle (in radians) Δt - the change in time (in seconds) ω needs to be in the units of rad/s ω = Δθ / Δt ω - angular velocity (the Greek letter omega) Δθ - the change in angle (in radians) Δt - the change in time (in seconds) ω needs to be in the units of rad/s

6. Example 1 A gymnast on a high bar swings through two revolutions in 1.90 s. Find the average angular velocity (in rad/s) of the gymnast.

7. Rotational Motion Angular velocity (ω) should not be confused with linear velocity (v) All points have the same angular velocity, but the farther away from the axis of rotation the greater the linear velocity Angular velocity (ω) should not be confused with linear velocity (v) All points have the same angular velocity, but the farther away from the axis of rotation the greater the linear velocity

8. Rotational Motion v = rω v - velocity (m/s) r - radius of circle (m) ω - angular velocity (rad/s) v = rω v - velocity (m/s) r - radius of circle (m) ω - angular velocity (rad/s)

9. Rotational Motion angular acceleration occurs when a rotating object speeds up or slows down its rate of rotation angular acceleration like angular velocity is constant throughout the rotating body angular acceleration occurs when a rotating object speeds up or slows down its rate of rotation angular acceleration like angular velocity is constant throughout the rotating body

10. Rotational Motion α = Δω / Δt α - angular acceleration (the Greek letter alpha) Δω - change is angular velocity (rad/s) Δt - change in time (s) α = Δω / Δt α - angular acceleration (the Greek letter alpha) Δω - change is angular velocity (rad/s) Δt - change in time (s)

11. Rotational Motion α is measured in rad/s 2 the acceleration is occurring tangential to the circle a tan = rα a = ω 2 r α is measured in rad/s 2 the acceleration is occurring tangential to the circle a tan = rα a = ω 2 r

12. Example 2 A jet awaiting clearance for takeoff is momentarily stopped on the runway. As seen from the front of one engine the fan blades are rotating with an angular velocity of -110 rad/s. As the plane takes off the blades reach an angular velocity of -330 rad/s in a time of 14 s. Find the angular acceleration.

13. Rotational Motion Frequency - the number of revolutions an object makes in one second Frequency (f) is measured in Hertz (Hz) ω = 2πf Frequency - the number of revolutions an object makes in one second Frequency (f) is measured in Hertz (Hz) ω = 2πf

14. Rotational Motion The reciprocal of frequency is period Period (T) - the time it takes for one revolution Period is measured in seconds T = 1/f The reciprocal of frequency is period Period (T) - the time it takes for one revolution Period is measured in seconds T = 1/f

15. Rotational Motion For linear motion we derived 4 equations for constant acceleration We can use the same procedures to derive the same equations for rotational motion For linear motion we derived 4 equations for constant acceleration We can use the same procedures to derive the same equations for rotational motion

16. Rotational Motion ω = ω o + αt θ = ω o t + (1/2)αt 2 ω 2 = ω o 2 + 2αθ ϖ = (ω + ω o ) / 2 ω = ω o + αt θ = ω o t + (1/2)αt 2 ω 2 = ω o 2 + 2αθ ϖ = (ω + ω o ) / 2

17. Example 3 A centrifuge starts from rest and accelerates uniformly at the rate of 10.0 rad/s 2. What is the angular velocity after 5 s?

18. Example 4 Using the data from example 3, how many revolutions will the centrifuge make in 6 s?

19. Example 5 A blender’s blades are rotating with an initial angular velocity of 375 rad/s. The speed setting in increased and the blades rotate through 44 rad. The angular acceleration is 1740 rad/s 2. Find the final angular velocity.