Rotational Motion

1. Rotational Motion Rotation about a fixed axis

2. Rotational Motion • Translation • Rotation • Rolling

3. Angular Kinematics Describing Angular Motion

4. Angular Position - q • Displacement is now defined as angle of rotation. • Polar coordinates (r, q) will make it easier to notate this motion. Dq

5. Angular Position - q • For a circle: 2p=circumference/radius • So: q = s/r or s = qr • s => arc length • r => radius • q => angle, measured in radians • Remember: 360º = 2p radians • Displacement: Dq = q1– q0

6. Angular Velocity - w • Average angular velocity • Instantaneous angular velocity

7. Angular Direction • Right-hand coordinate system • w is positive when q is increasing, which happens in the counter-clockwise direction • w is negative when q is decreasing, which happens in the clockwise direction

8. Angular Acceleration - a • Average Angular Acceleration • Instantaneous Angular Acceleration

9. example 1 q = (t3 - 27t + 4)rad • w = ? • a = ? • At what time(s) does w = 0 rad/s?

10. example 2 a = (5t3 – 4t)rad/s2 At t = 0s: w = 5 rad/s and q = 2 rad. • w= ? • q = ?

11. Linear to Angular Conversions • Position: s = qr • Velocity: v = wr • Acceleration: a = ar • Remember, also, centripetal accleration: ac = v2/r =w2r

12. Angular Kinematics*constant a

13. example 3 A turntable starts from rest and begins rotating in a clockwise direction. 10 seconds later, it is rotating at 33.3 revolutions per minute. • What is the final angular velocity (rad/s)? • What is the average angular acceleration? • How far did a point 10 cm from the center travel in that 10 seconds, both angularly and linearly?

14. example 4 In an astronaut training centrifuge (r = 15m): • What constant w would give 11g’s? • How fast is this in terms of linear speed? • What is the translational acceleration to get to this speed from rest in 2 minutes?

15. Torque The tendency of a force to cause angular motion

16. Torque • Torque is dependent on the amount and location of the force applied to an object. • Where r is the distance between the pivot point and the force and q is the angle between r and F.

17. example 5 • A one piece cylinder has a core section that protrudes from a larger drum. A rope wrapped around the large drum of radius, R, exerts a force, F1, to the right, while a rope wrapped around the core, radius r, exerts a force, F2 downward.. • Calculate the net torque, in variables. • If F1=5 N, R = 1 m, F2=6 N, and r = 0.5 m, calculate the net torque.

18. Cross Product The “other” vector multiplication

19. Cross product • Results in a vector quantity • Calculates the perpendicular product of two vectors • The product of any two parallel vectors will always be zero.

20. Point your fingers along the radius Curl them in the direction of the force Your thumb will be pointing in the direction of the rotation. http://hyperphysics.phy-astr.gsu.edu/hbase/tord.html Rotational Direction is defined by the axis that it rotates around. The Right Hand Rule

21. Cross Product

22. Calculating Cross Product

23. Finding the Determinant

24. example 6 • Find the cross product of 7i + 5j – 3k and 2i-8j + 7k. • What is the angle between these two vectors?

25. example 7 • A plumber slips a piece of scrap pipe over his wrench handle to help loosing a pipe fitting. He then applies his full weight (900 N) to the end of the pipe by standing on it. The distance from the fitting to his foot is 0.8 m, and the wrench and pipe make a 19º angle with the ground. Find the magnitude and direction of the torque being applied.

26. example 7 continued… r = 0.8m q = 19º F = 900 N

27. example 8 One force acting on a machine part is F = (-5i + 4j)N. The vector from the origin to the point applied is r = (-0.5i + 0.2j)m. • Sketch r and F with respect to the origin • Determine the direction of the force with the right hand rule. • Calculate the torque produced by this force. • Verify that your direction agrees with your calculation.