Rotational Dynamics

1. Rotational Dynamics Chapter 9

2. Pure translational motion Pure rotational motion Combination of translational and rotational motions

3. According to Newton’s 2nd law ... A net force causes an object's velocity to change. Force is an interaction that can cause an object's translational velocity to change. Torque is an interaction that can cause an object's angular velocity to change. Forces produce torques.

4. Hinge Axis The amount of torque depends on ... 1) location of the axis of rotation 2) distance from the axis to the line of action 3) magnitude of the force The distance from the axis to the line of action is called the lever arm.

5. Torque = (Force) x (lever arm) Torque Torque is a vector quantity (torque use vector addition rules). Torque directionis positive when the force tends to produce a counter-clockwise rotation about the axis. Units: newton meter (N m) [ For torque, N m is not joules.]

6. Lever arm is the shortest distance from the axis to the line of action of the force. A force direction perpendicular to the lever arm produces maximum torque.

7. Lever arm

8. A force direction straight toward or straight away from the axis of rotation produces zero torque because the lever arm distance is zero.

9. Example 2 Achilles Tendon Tendon exerts a 720 N force on the heel. The tendon is attached 3.6 cm from the center of the ankle joint. Determine the magnitude and direction of the torque produced by the tendon force about an axis of rotation through the ankle joint. Lever arm Lever arm = (0.036 m) cos 55°= 0.02065 m Torque direction is clockwise (negative).

10. Airplane force produces tangent acceleration Lever arm Torque causes angular acceleration: particle Newton’s 2nd law After making the substitutions… Therefore For a single object of mass m moving a distance r from the center of a circle.

11. Net torque is the vector sum of the torques acting on a solid object. A solid object is made up of many particles.All particles have same angular acceleration α . Each particle has its own ... Distance r from the center Mass m Torque causes angular acceleration: solid object For one torque and one particle for a solid object

12. Moment of inertia for a solid object for a solid object is called the object's moment of inertiaI. Units: kg m2 Moment of inertia is an object's resistance to changing its rotational velocity. Moving the mass farther from the center of rotation increases the moment of inertial.

13. solid solid hollow Moment of inertia for various rigid objects of mass M.

14. moment of inertia Newton's 2nd law for rotational motion net torque Newton's second law for rotational motion for a solid object α is the object's angular acceleration

15. Objects with more inertial speed up more slowly Identical masses and identical diameters The hollow cylinder has its mass farther from the center so it has more inertia and gains velocity more slowly. The solid cylinder has its mass closer to the center so it has less inertia and gains velocity more rapidly.

16. arc length s torque Mechanical work for rotational motion Work equation for rotational motion Angle θmust be expressed in radians. SI Unit of Rotational Work: joule (J)

17. For many small particles or a rigid object rotating with angular velocity ω Kinetic energy for rotational motion ω For a small particle Angular velocity ωmust be in rad/s. SI Unit of Rotational Kinetic Energy: joule (J)

18. Total mechanical energy E = KE + PE Translational motion of the center and rotational motion around the center both have kinetic energy. Work done by non-conservative forces changes the total mechanical energy Conservation of total mechanical energy

19. Example 13 Rolling Cylinders A hollow cylinder and a solid cylinder start from rest at the top of a ramp. Determine the velocities at the bottom. Critical thinking step: Only gravity does any work. The support force always does zero work and there is no sliding displacement motion against the friction force. Therefore, the total mechanical energy is conserved.ΔE=0 and Efinal=Einitial.

20. For rolling motion 0 0 0 Since there is no work done by non-conservative forces the total mechanical energy is conserved. The cylinder with the bigger moment of inertia will have a smaller final translational velocity.

21. Units: Momentum momentum = (inertia) x (velocity) Review: Linear momentum vector Linear momentum = (linear inertia m) x (linear velocity v) Angular momentum = (moment of inertia I) (angular velocity ω) Angular velocity must be in rad/s.

22. Conservation of angular momentum Only a net external torque can change the total angular momentum of a system. Internal torques transfer angular momentum inside the system, but don't change the system's total angular momentum. Therefore, if the net external torque is equal to zero, the total angular momentum of the system stays the same. Total angular momentum of a system is conserved when the net external torque is equal to zero.

23. Conceptual Example 14 A Spinning Skater no off-axis forces An ice skater is spinning with her arms and a leg stretched out. She pulls her arms and leg inward and her spinning motion speeds up dramatically. Use the principle of conservation of angular momentum to explain why her spinning motion changes. Moving mass inward reduces I which causes ωto increase.

24. Example 15 A Satellite in an Elliptical Orbit A satellite moves in an elliptical orbit about the Earth. At its closest distance (perigee) it is 8.37x106 m from the center of the Earth. At its greatest distance (apogee) it is 25.1x106 m from the center of the Earth. The satellite's velocity the perigee is 8450 m/s. Find the velocity at the apogee. Critical thinking steps: What rule should we use? Kinematics, energy, momentum, …. ???

25. Force Critical thinking steps: The gravity force depends on distance, and since the distance changes, the gravity force changes so we can’t use equations for constant accelerated motion. The only force on the satellite is gravity so the total mechanical energy is conserved, but we only have an equation for potential energy for objects near the surface of the earth. Does gravity produce a torque on the satellite's motion? The gravity force pulls toward the center of the Earth and the axis of rotation for the orbit is the center of the Earth. A force directed straight toward the axis of rotation has zero lever arm and therefore produces zero torque. Since there is no external torques acting on it, the satellite's total angular momentum is conserved.

26. Use conservation of angular momentum For a particle For a tangent velocity The satellite moves slower when it is farther from the Earth.

27. The End