Rotational Dynamics

1. Rotational Dynamics Chapter 8 Section 3

2. Torque Direction • A net positive torque causes an object to rotate counterclockwise. • A net negative torque causes an object to rotate clockwise. + -

3. Newton’s Second Law for Rotation ΣT = Iα Net Torque = (Moment of Inertia)(Angular Acceleration)

4. Translational vs. RotationalNewton’s 2nd Law Translational:F = ma Force = Mass x Acceleration Rotational: T = Iα Torque = Moment of Inertia x Angular Acceleration

5. Example Problem #1 • A toy flying disk with a mass of 165.0 grams and a radius of 13.5 cm that is spinning at 30 rad/s can be stopped by a hand in 0.10 sec. What is the average torque exerted on the disk by the hand?

6. Example Problem #1 Answer T = Iα T = (½mr2)((ωf-ωi)/t) T = (½)(0.165kg)(0.135m)2 ((0rad/s – 30rad/s)/0.10s) T = -0.45 Nm

7. Resistance to Change • Swinging a sledge hammer, or a similarly heavy object, takes some effort to start rotating the object. • The same can be said about stopping a heavy object that is rotating.

8. Momentum • Translational Momentum – A vector quantity defined as the product of an object’s mass and velocity. • Also known as, “Inertia In Motion” • Angular Momentum – The product of a rotating object’s moment of inertia and angular speed about the same axis.

9. Angular Momentum Equation L = Iω Angular Momentum = (Moment of Inertia)(Angular Speed)

10. Angular Momentum • The variable used for Angular Momentum. • Capital letter “L” • The SI units for angular momentum. • Kgm2/s

11. Translational vs. RotationalMomentum Translational:p= mv Linear Momentum = mass x velocity Rotational: L = Iω Angular Momentum = moment of inertia x angular speed

12. Conservation of Angular Momentum • The Law of Conservation of Angular Momentum - When the net external torque acting on an object is zero, the angular momentum of the object does not change. Li = Lf Iωi = Iωf

13. Angular Momentum Example • Angular momentum is conserved as a skater pulls his arms towards their body, assuming the ice they are skating on is frictionless. • During an ice skaters spin, they will bring their hands and feet closer to the body which will in turn decrease the moment of inertia and as a result increase the angular speed.

14. Example Problem #2 • A 0.11kg mouse rides on the edge of a rotating disk that has a mass of 1.3 kg and a radius of 0.25m. If the rotating disk begins with an initial angular speed of 3.0 rad/s, what is its angular speed after the mouse walks from the edge to a point 0.15m from the center? What is the tangential speed of the disk at the outer edge?

15. Example Problem #2 Answer • ω = 3.2 rad/s • v = 0.8 m/s

16. Kinetic Energy • Rotational Kinetic Energy – Energy of an object due to its rotational motion. • Greater angular speeds and greater moment of Inertia, yields greater rotational kinetic energy

17. Rotational Kinetic Energy Equation Rotational Kinetic Energy = ½(Moment of Inertia) (Angular Speed)2

18. Momentum vs. Energy • Unlike Angular momentum, rotational energy increases when the moment of inertia deceases when no external torques are introduced. • A greater angular speed will increase rotational kinetic energy because of the square term in the equation.

19. Example Problem #3 • A car tire has a diameter of 0.89m and may be approximated as a hoop. How fast will it be going starting from rest to roll without slipping 4.0m down an incline that makes an angle of 35 degrees with the horizontal?

20. Example Problem #3 Diagram Vi = 0 m/s 0.89m h = ? d = 4.0m Vf = ?

21. Example Problem #3 Answer • Remember that mechanical energy within a system must remain constant.