Rotational Dynamics

Chapter 9

1.         What is Torque

            a.         Combination of force and distance from axis

            b.         Produces rotational motion in rigid object

            c.         Line of action: direction of force

            d.         Lever arm: distance from axis to line of action

            e.         Formula: t = Fl

            f.         Unit: newton•meter

            g.         Positive: counterclockwise rotation

            h.         Negative: clockwise rotation

2.         Example 1 (page 239)

            a.         Shows effect of lever arm

            b.         Set up demo with pulleys and weights

3.         Example 2 (page 239-240)

            a.         Shows effect of angle

4.         Equilibrium

            a.         Body is rigid

            b.         Zero translational acceleration

            c.         Zero angular acceleration

            d.         Sum of external forces zero

            e.         Sum of external torques zero

5.         Problem solving

            a.         Select object

            b.         Draw free body diagram showing external forces

            c.         Chose a set of axes (x.y) and resolve forces

            d.         Apply the conditions that balance the forces

            e.         Select a convenient axis and calculate the torques

            f.         Set sum of torques to zero

            g.         Solve equations d and f for unknown variables

6.         PS Note 1

            a.         The location of the axis is arbitrary since system is at equilibrium

            b.         Example 3 (page 242)

7.         PS Note 2

            a.         If the chosen direction is backward in the free body diagram the sign of the force will be negative

            b.         Example 5 (page 244)

8.         PS Note 3

            a.         It may be necessary to chose different objects/points for analysis at different stages of solving a problems

            b.         Example 6 (page 245-246)

9.         Center of Gravity

            a.         Point in a rigid body where weight acts when calculating torque

            b.         Symmetric objects have center of gravity at geometric center

            c.         System of objects (example 7, page 248)

                        i.         Combine torques and solve for x_cg

                        ii.        W₁x₁ + W₂x₂ = (W₁+ W₂) x_cg

                        iii.       Separate term for each object in system

            d.         Irregular Object

                        i.         Suspend from two points

                        ii.        Center of gravity is at intersection of plumb lines

10.       Example 8 (page 248-249)

            a.         Link to NASCAR and towing

            b.         Link to farm tractors

11.       Second Law and Torque

            a.         Torque is proportional to angular acceleration

            b.         Proportionality constant is moment of inertia (I)

            c.         I depends on:

                        i.         Location and orientation of axis

                        ii.        Shape of object

                        iii.       See table 9.1, page 253

            d.         Formula: St = Ia

            e.         a must be in rad/s²

12.       Rotational Work

            a.         Torque turning object through angle

            b.         Formula: W_R = tq

            c.         q must be in radians

            d.         Unit: Joule

13.       Rotational Energy (example 13, page 259)

            a.         Based on rotational velocity (w)

            b.         Formula: KE_R = ½ Iw²

            c.         w must be in rad/s

            d.         Unit: Joule

            e.         E_T = ½ mv² + ½ Iw² + mgh

14.       Angular momentum

            a.         Based on rotational velocity

            b.         Formula: L = Iw

            c.         w must be in rad/s

            d.         Unit: kg•m²/s

            e.         Conserved when external torques sum to zero

            f.         Example 14: link to effect of change in I

            g.         Example 15: link to Kepler