1. angle of rotation (rads) angular velocity (rad/s) angular acceleration (rad/s 2 ) Recall that centripetal acceleration is expressed in terms of tangential velocity as: a r = v 2 /r. How is it expressed in terms of angular velocity ω? A) a r = ω 2 /rB) a r = rω C) a r = rω 2 D) a r = r 2 ω 2 a r = v 2 /r = (rω) 2 /r = rω 2 Clicker QuestionRoom Frequency BA

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

4. constant angle of rotation (rads) angular velocity (rad/s) angular acceleration (rad/s 2 ) Constant acceleration a:Constant angular acceleration α: Rotational Kinematics

5. A space ship is initially rotating on its axis with an angular velocity of ω 0 = 0.5 rad/sec. The Captain creates a maneuver that produces a constant angular acceleration of α = 0.1 rad/sec 2 for 10 sec. 1) What is its angular velocity at the end of this maneuver? A)1 rad/secB) 1.5 rad/sec C) 0.6 rad/secD) 5 rad/sec Clicker QuestionRoom Frequency BA

6. A space ship is initially rotating on its axis with an angular velocity of ω 0 = 0.5 rad/sec. The Captain creates a maneuver that produces a constant angular acceleration of α = 0.1 rad/sec 2 for 10 sec. 1) What is its angular velocity at the end of this maneuver? Clicker QuestionRoom Frequency BA 2) At how many rpms is it rotating after the maneuver? A) 10 rpmB) (30/π) rpmC) (45/π) rpmD) π rpm

7. 3) How far (in radians) did the space ship rotate while the Captain was applying the angular acceleration? A) 1 radB) 5 radC) 10 radD) 45 rad Clicker QuestionRoom Frequency BA A space ship is initially rotating on its axis with an angular velocity of ω 0 = 0.5 rad/sec. The Captain creates a maneuver that produces a constant angular acceleration of α = 0.1 rad/sec 2 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?

8. 4) How many degrees is this? A)180°B) 1800° C) (1800/π) degreesD) 360° Clicker QuestionRoom Frequency BA A space ship is initially rotating on its axis with an angular velocity of ω 0 = 0.5 rad/sec. The Captain creates a maneuver that produces a constant angular acceleration of α = 0.1 rad/sec 2 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?

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

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

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

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

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

14. What’s the torque you exerted on the door? (Note: sin 30° =1/2) A)8 NmB) 10 Nm C) 12 NmD) 20 Nm Clicker QuestionRoom 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.

15. 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? Clicker QuestionRoom Frequency BA Torque

16. 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? A)0.707 LFB) -0.707 LFC) 1.707 LFD) -1.707 LF E) Zero Clicker QuestionRoom Frequency BA Torque

17. Torque and Moment of Inertia Rotational Analog to Newton’s Second Law (F = ma): 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. Units of τ: Nm Units of I : kg m 2 Units of α: rad/s 2 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. m r r Thus, the moment of inertia for a point mass is equal to mr 2.

19. 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) mL 2 B) 2mL 2 C) 4mL 2 D) 5mL 2 E) 9mL 2 up down Clicker QuestionRoom Frequency BA

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