Rotational Motion: Understanding Circular Motion and Torque

1. Chapter 8 Rotational Motion

2. Circular Motion • Specific type of 2-dimensional motion. • In discussing circular motion: • Angular speed • Angular acceleration • Centripetal acceleration

3. Linear Vs. Angular

4. Angular Displacement:∆θ • Axis of rotation is the center of the disk. • Need a fixed reference line. • During time t, the reference line moves through angle θ.

5. Average Angular Speed: ω • The average angular speed, ω, of a rotating rigid object is the ratio of the angular ___________ to the __________.

6. Average Angular Acceleration: α • The average angular acceleration of an object is defined as the ratio of the change in the __________ to the _____ it takes for the object to undergo the change:

7. Sample Problem If a truck has a linear acceleration of 1.85m/s2 and the wheels have an angular acceleration of 5.23 rad/s2, what is the diameter of the truck’s wheels?

8. Linear Vs. Angular – Part 2 • When a rigid object rotates about a fixed axis every portion of the object has the _____________and __________. This is not true with the linear speed and acceleration.

9. Displacements Speeds Accelerations Every point on the rotating object has the same angular motion. Every point on the rotating object does not have the same linear motion. Relationship Between Angular and Linear Quantities

10. Sample Problem A wheel rotates with a constant angular acceleration of 3.50 rad/s2. If the angular speed of the wheel is 2.00 rad/s at t= 0s, through what angle does the wheel rotate between t =0 and t=2.00 s?

11. Vector Nature of Angular Quantities • Angular displacement, velocity and acceleration are all vector quantities. • Direction can be more completely defined by using the right hand rule. • Grasp the axis of rotation with your right hand. • Wrap your fingers in the direction of rotation. • Your thumb points in the direction of ω.

12. Velocity Directions, Example • In a, the disk rotates clockwise, the velocity is into the page • In b, the disk rotates counterclockwise, the velocity is out of the page

13. Force vs. Torque • Forces cause ___________ accelerations. • Torques cause ____________ accelerations. • Force and torque are related. • Think of torque as the rotational equivalent to force.

14. Torque and Newton’s 2nd Law • Translational Motion • Rotational Motion • I = moment of inertia • = angular acceleration

15. Torque • The door is free to rotate about an axis through O. • There are three factors that determine the effectiveness of the force in opening the door: • The magnitude of the force. • The position of the application of the force. • The angle at which the force is applied.

16. Torque • Torque, t, is the ________ of a force to _______ an object about some ____. • t = r F • t is the torque • Symbol is the Greek tau • F is the force • r is the length of the position vector • SI unit is N.m

17. Torque • For a given force, torque increases with increasing distance from the axis of rotation (r). • t = r F

18. Torque • For a given force, torque is at a ________ when the angle between the force and r is at ___ and at a ______ when the angle is __________. • t = r F

19. Direction of Torque • Torque is a vector quantity • The direction is ___________ to the plane determined by the position vector and the force. • If the turning tendency of the force is ______________, the torque will be __________. • If the turning tendency is ________, the torque will be ________.

20. Multiple Torques • When two or more torques are acting on an object, the torques are ________. • This is a vector addition. • If the net torque is zero, the object’s rate of rotation doesn’t change.

21. Sample Problem A packing crate is being pushed by two forces of equal magnitude acting in opposite directions. Find the net torque exerted on the crate by these two forces have a magnitude of 500 N and the width of the crate is 1.0 m. Assume the axis of rotation is through the center of the crate.

22. Lever Arm • The lever arm, d, is the ___________ distance from the axis of rotation to a line drawn along the direction of the force • d = r sin q • This also gives t = F r sin q

23. Sample Problem Find the torque produced by the 300 N force applied at an angle of 60o to the door of the figure at the right.

24. Right Hand Rule • Point the fingers in the direction of the position vector • Curl the fingers toward the force vector • The thumb points in the direction of the torque

25. Torque and Equilibrium • First Condition of Equilibrium • The net external force must be zero: • This is a necessary, but not sufficient, condition to ensure that an object is in complete mechanical equilibrium. • This is a statement of translational equilibrium.

26. Torque and Equilibrium • To ensure ____________ equilibrium, you need to ensure _________ equilibrium as well as ___________. • The Second Condition of Equilibrium states: • The ___________________must be zero.

27. If an object has the ability to rotate, but is not moving at all, i.e. F = 0 and  = 0 Then the object is in STATIC EQUILIBRIUM. Applications: Seesaws: Consider a seesaw that is not moving, has 2 people on it and is supported off centre.

28. Sample Problem Little Johnny (40kg) and Silly Sally (30kg) are playing on a 3.0m long seesaw (60kg) that is supported in the middle. If Silly Sally is sitting on the very edge, how far from the centre should little Johnny sit in order for the seesaw to balance horizontally?

29. Notes About Equilibrium • A zero net torque does not mean the _________ of rotational motion. • An object that rotates at uniform angular velocity can be under the influence of a zero net torque. • This is analogous to the translational situation where a zero net force does not mean the object is not in motion.

30. THE END Chapter 8 Rotational Motion