angle of rotation ( rads )

1. Clicker Question Room Frequency BA angle of rotation (rads) angular velocity (rad/s) angular acceleration (rad/s2) Recall that centripetal acceleration is expressed in terms of tangential velocity as: ar = v2/r. How is it expressed in terms of angular velocity ω? A) ar = ω2/r B) ar = rω C) ar = rω2 D) ar = r2ω2 ar= v2/r = (rω)2/r = rω2

2. Remember: This Week • Read Ch. 8: And read Dr. Dubson’s excellent notes on Ch. 8. • CAPA assignment #10 is due this Friday at 10 PM. • Recitation: Review 1 • Lab Make-up (Labs 1-3): arrange with your TA.

3. Rotational Kinematics angle of rotation (rads) angular velocity (rad/s) angular acceleration (rad/s2) Today: torque (N m) moment of inertia (kg m2) Newton’s 2nd Law IC (Conservation of Mechanical Energy)

4. Rotational Kinematics angle of rotation (rads) angular velocity (rad/s) angular acceleration (rad/s2) constant Constant acceleration a: Constant angular acceleration α:

5. Clicker Question Room Frequency BA A space ship is initially rotating on its axis with an angular velocity of ω0 = 0.5 rad/sec. The Captain creates a maneuver that produces a constant angular acceleration of α = 0.1 rad/sec2 for 10 sec. 1) What is its angular velocity at the end of this maneuver? 1 rad/sec B) 1.5 rad/sec C) 0.6 rad/sec D) 5 rad/sec

6. Clicker Question Room Frequency BA A space ship is initially rotating on its axis with an angular velocity of ω0 = 0.5 rad/sec. The Captain creates a maneuver that produces a constant angular acceleration of α = 0.1 rad/sec2 for 10 sec. 1) What is its angular velocity at the end of this maneuver? 2) At how many rpms is it rotating after the maneuver? A) 10 rpm B) (30/π) rpm C) (45/π) rpm D) π rpm

7. Clicker Question Room Frequency BA A space ship is initially rotating on its axis with an angular velocity of ω0 = 0.5 rad/sec. The Captain creates a maneuver that produces a constant angular acceleration of α = 0.1 rad/sec2 for 10 sec. 1) What is its angular velocity at the end of this maneuver? 2) At how many rpms is it rotating after the maneuver? 3) How far (in radians) did the space ship rotate while the Captain was applying the angular acceleration? A) 1 rad B) 5 rad C) 10 rad D) 45 rad

8. Clicker Question Room Frequency BA A space ship is initially rotating on its axis with an angular velocity of ω0 = 0.5 rad/sec. The Captain creates a maneuver that produces a constant angular acceleration of α = 0.1 rad/sec2 for 10 sec. 1) What is its angular velocity at the end of this maneuver? 2) At how many rpms is it rotating after the maneuver? 3) How far (in radians) did the space ship rotate while the Captain was applying the angular acceleration? 4) How many degrees is this? 180° B) 1800° C) (1800/π) degrees D) 360°

9. Rotational Dynamics angle of rotation (rads) angular velocity (rad/s) angular acceleration (rad/s2) Today: torque (N m) moment of inertia (kg m2) Newton’s 2nd Law IC

10. Rotational Dynamics Torque Consider the work done by a constant force in moving an object from point A to point B on a circle. r s θ A r (Component of force along displacement) x displacement Torque Units: N m

11. Torque The amount of work done by increases linearly with r and . Torque: = (lever arm) x (Component of force perpendicular to lever arm)

12. Torque Therefore, to easily rotate an object about an axis, you want a large lever arm r and a large perpendicular force :

13. Torque Therefore, to easily rotate an object about an axis, you want a large lever arm r and a large perpendicular force :

14. Clicker Question Room Frequency BA Torque You pull on a door handle a distance r = 1m from the hinge with a force of magnitude 20 N at an angle θ =30° from the plane of the door. What’s the torque you exerted on the door? (Note: sin 30° =1/2) 8 Nm B) 10 Nm C) 12 Nm D) 20 Nm

15. Clicker Question Room Frequency BA Torque Three forces labeled A, B, and C are applied to a rod which pivots on an axis through its center. Which force causes the largest size torque?

16. Clicker Question Room Frequency BA Torque Three forces labeled A, B, and C are applied to a rod which pivots on an axis through its center. What is the net torque on the rod? 0.707 LF B) -0.707 LF C) 1.707 LF D) -1.707 LF E) Zero

17. Torque and Moment of Inertia Rotational Analog to Newton’s Second Law (F = ma): Units of τ: Nm Units of I: kg m2 Units of α: rad/s2 Torque = (moment of inertia) x (rotational acceleration) Moment of inertia I depends on the distribution of mass and, therefore, on the shape of an object. Moment of inertia for a point mass is the mass times the square of the distance of the mass from the axis of rotation.

18. Torque and Moment of Inertia Consider a point mass m to which we apply a constant tangential force . r r m Thus, the moment of inertia for a point mass is equal to mr2.

19. Clicker Question Room Frequency BA up down A light rod of length 2L has two heavy masses (each with mass m) attached at the end and middle. The axis of rotation is at one end. What is the moment of inertia about the axis? A) mL2 B) 2mL2 C) 4mL2 D) 5mL2 E) 9mL2

20. Rotational Dynamics angle of rotation (rads) angular velocity (rad/s) angular acceleration (rad/s2) torque (N m) moment of inertia (kg m2) Newton’s 2nd Law Kinetic Energy (joules J) IC (Conservation of Mechanical Energy) Tod