Angular Kinematics: Rotational Motion

1. TOPIC 4 Angular Kinematics: Rotational Motion

2. Lecture Objectives • After this lecture, students will be able to: • explain angular motion • to define radian, angular velocity and angular acceleration • to explain relationship between straight line motion and angular motion.

3. Linear motion describes a point • Rotational motion: object is defined by a minimum of 2 points Displacement = 7m Displacement = 30 deg

4. Planes of Motion Axes of Rotation (Internal or External)

5. q r r s RADIANS (q) and Revolutions Angular displacement • for a full circle, θ = 360° = 1 rev = 2π radian • 1 rad = 360°/ 2p = 57.3 ° = 0.16 rev • radians are dimensionless (ratio of a length divided by a length) • Formula for length of arc, s = r θ

6. D q w average= D t 3. Angular Velocity (w) • How fast the body is changing it angular position. • SI unit for angular velocity is “ rad/s “ • Angular velocity is positive when the rotation is counterclockwise EXAMPLE : • What is the angular velocity (rad/s) of a gymnast on a high bar swings through 2 revolutions in clockwise direction in 1.9 seconds? Solution: Angular displacement ∆θ = 2rev X - 2 = - 4 therefore :

7. 4. Angular Acceleration () • The rate at which the body’s angular velocity (w) changes • The SI unit for angular acceleration is Example : A jet fan blades are rotating with angular velocity of -100 rad/s. As the jet takes off the angular velocity of the blades reaches -330 rad/s in a time of 14 s. Find the angular acceleration assuming it to be constant. Solution :

8. COMPARISON Equations of Rotational Kinematics Equations of Linear Motion The form of the rotational equations is exactly the same as the linear ones we studied earlier and the handling is just the same. You only need to remember that w must be in radians per second (r/s) and a in r/s2.

9. Question 1: An airliner arrives at the terminal and the engines are shut off.  The rotor of one of the engines has an initial clockwise angular speed of 2000 rad/s.  The engines rotation slows with with an angular acceleration of magnitude 80 rad/s2. (a) Determine the angular speed after 10s. (b) How long does it take the rotor to come to rest?

11. Question 2: A metal cylinder of radius 0.45 meters is spinning at 2000 rpm and a brake is applied slowing it to 1000 rpm in 10 seconds. A)what is the angular acceleration? B)how many revolutions does it make before it reaches 1000 rpm? C)how far does a point on the edge of the disk travel during this time?

12. Answer: The first thing to do is convert 2000 rpm (revolutions per minute) and 1000 rpm into radians per second: In one revolution there are 2 π radians (or 360 degrees) In one minute there are sixty seconds So,  (2000 rev/min) x (2 πrad/rev) x (1 min/60 sec) = 209.4 r/sec And (1000 rev/min) x (2 πrad/rev) x (1 min/60 sec) = 104.7 r/sec a) ω f = ω i + α t (104.7 r/s) = (209.4 r/s) + α(10 s) α = -10.5 r/s2 b) θ = ω it + (1/2) α t2 = (209.4 r/s)(10 s) + (1/2)(-10.5 r/s2)(10 s)2 = 1569 radians We must convert this to the number of rotations! Each rotation contains 2 π radians, so 1569/(2 π) = 250 rotations

13. c) Since we're talking about distance in meters then we need to calculate how many meters we move in one rotation. Then we can find out how many meters are in 250 rotations: Meters in one rotation = 2 π r (circumference) = 2.83 meters So the total distance traveled = (250 rotations)(2.83 meters/rotation) = 707.5 meters.