Torque and Equilibrium in Rotational Dynamics

1. Chapter 9 Rotational Dynamics

2. 9.1 The Action of Forces and Torques on Rigid Objects In pure translational motion, all points on an object travel on parallel paths. The most general motion is a combination of translation and rotation.

3. 9.1 The Action of Forces and Torques on Rigid Objects According to Newton’s second law, a net force causes an object to have an acceleration. What causes an object to have an angular acceleration? TORQUE

4. 9.1 The Action of Forces and Torques on Rigid Objects The amount of torque depends on where and in what direction the force is applied, as well as the location of the axis of rotation.

5. 9.1 The Action of Forces and Torques on Rigid Objects DEFINITION OF TORQUE Magnitude of Torque = (Magnitude of the force) x (Lever arm) Direction: The torque is positive when the force tends to produce a counterclockwise rotation about the axis. SI Unit of Torque: newton x meter (N·m)

6. 9.1 The Action of Forces and Torques on Rigid Objects Example 2 The Achilles Tendon The tendon exerts a force of magnitude 790 N. Determine the torque (magnitude and direction) of this force about the ankle joint.

7. 9.1 The Action of Forces and Torques on Rigid Objects 790 N

8. 9.1 Problems on Action of Forces and Torques on Rigid Objects Problem 1: If, the torque on a shaft of radius 2.37 cm is 38 Nm, what force is applied to the shaft? Problem 2: A motorcycle head bolt is torqued to 25 Nm. What length shaft do we need on a wrench to exert a maximum force of 70 N? Problem 3: A truck mechanic must loosen a rusted lug nut. If the torque required to loosen the nut is 60 Nm, what force must be applied to a 35 cm wrench?

9. 9.2 Rigid Objects in Equilibrium If a rigid body is in equilibrium, neither its linear motion nor its rotational motion changes.

10. 9.2 Rigid Objects in Equilibrium EQUILIBRIUM OF A RIGID BODY A rigid body is in equilibrium if it has zero translational acceleration and zero angular acceleration. In equilibrium, The sum of the externally applied forces is zero. (The sum of all parallel forces on a body in equilibrium must be zero) The sum of the externally applied torques is zero. (The sum of clockwise torques on a body in equilibrium must equal the sum of counterclockwise torques about any point).

11. 9.2 Rigid Objects in Equilibrium • Reasoning Strategy • Select the object to which the equations for equilibrium are to be applied. • 2. Draw a free-body diagram that shows all of the external forces acting on the • object. • Choose a convenient set of x, y axes and resolve all forces into components • that lie along these axes. • Apply the equations that specify the balance of forces at equilibrium. (Set the • net force in the x and y directions equal to zero.) • Select a convenient axis of rotation. Set the sum of the torques about this • axis equal to zero. • 6. Solve the equations for the desired unknown quantities.

12. 9.2 Rigid Objects in Equilibrium • Steps to solve parallel force problems: • Sketch the problem. • 2. Write an equation setting the sums of the opposite forces to each other. • Choose a point of rotation. Eliminate a variable, if possible by making its torque • arm zero. • Write the sum of all clockwise torques. • 5. Write the sum of all counterclockwise torques. • Set clockwise torque equal counter clockwise torque. • Solve the equation for sum of all clockwise torques equal sum of all • counterclockwise torque for the unknown quantity. • Substitute the value found in step 7 into the equation in step 2 to find the other • unknown quantity.

13. 9.2 Rigid Objects in Equilibrium Example 3 A Diving Board A woman whose weight is 530 N is poised at the right end of a diving board with length 3.90 m. The board has negligible weight and is supported by a fulcrum 1.40 m away from the left end. Find the forces that the bolt and the fulcrum exert on the board.

14. 9.2 Rigid Objects in Equilibrium

15. 9.2 Rigid Objects in Equilibrium

16. 9.2 Problems on Rigid Objects in Equilibrium Problem 4: A 2400 kg truck is 6 m from one end of a 27 m long bridge. A 1500 kg car is 10 m from the same end. How much weight must each end of the bridge support? Problem 5: Two painters, one of mass 75 kg, and the other 90 kg, stand on a 12 m scaffold, 6 m apart and 3 m from each end. They share a paint container of mass 21 kg in the middle of the scaffold. What weight must be supported by each of the ropes secured at the end of the scaffold? Problem 6: An auto differential with mass of 76 kg is 1 m from the end of a 2.2 m workbench. What mass must each end of the workbench support?

17. 9.3 Center of Gravity DEFINITION OF CENTER OF GRAVITY The center of gravity of a rigid body is the point at which its weight can be considered to act when the torque due to the weight is being calculated.

18. 9.3 Center of Gravity When an object has a symmetrical shape and its weight is distributed uniformly, the center of gravity lies at its geometrical center.

19. 9.3 Center of Gravity

20. 9.3 Center of Gravity Example 6 The Center of Gravity of an Arm The horizontal arm is composed of three parts: the upper arm (17 N), the lower arm (11 N), and the hand (4.2 N). Find the center of gravity of the arm relative to the shoulder joint.

21. 9.3 Center of Gravity

22. 9.3 Center of Gravity Conceptual Example 7 Overloading a Cargo Plane This accident occurred because the plane was overloaded toward the rear. How did a shift in the center of gravity of the plane cause the accident?

23. 9.3 Center of Gravity Finding the center of gravity of an irregular shape.

24. 9.3Problems on Equilibrium Involving Center of Gravity Problem 7: A bridge across a country stream weighs 89,200 N. A large truck stalls 4 m from one end of the 9 m bridge. What weight must each of the piers support if the truck weighs 98,000 N? Problem 8: A sign shown in (figure to be produced on the board) is 4 m long and weighs 1550 N, and is made of uniform material. A weight of 245 N hangs 1 m from the end. Find the tension in each support cable. Problem 9 (Homework. Due April 14, 2018): Juan and Pablo carry a load weighing 720 N on a pole between them. If the pole is 2 m long and the load is 0.5 m from Pablo, what force does each person support? Neglect the weight of the pole. If the weight of the 120 N pole is considered, what force does each person support?

25. 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis Moment of Inertia, I

26. 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis Net external torque Moment of inertia

27. 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis ROTATIONAL ANALOG OF NEWTON’S SECOND LAW FOR A RIGID BODY ROTATING ABOUT A FIXED AXIS Requirement: Angular acceleration must be expressed in radians/s2.

28. 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis Example 9 The Moment of Inertia Depends on Where the Axis Is. Two particles each have mass m and are fixed at the ends of a thin rigid rod. The length of the rod is L. Find the moment of inertia when this object rotates relative to an axis that is perpendicular to the rod at (a) one end and (b) the center.

29. 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis (a)

30. 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis (b)

31. 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis

32. 9.5 Rotational Work and Energy

33. 9.5 Rotational Work and Energy DEFINITION OF ROTATIONAL WORK The rotational work done by a constant torque in turning an object through an angle is Requirement: The angle must be expressed in radians. SI Unit of Rotational Work: joule (J)

34. 9.5 Rotational Work and Energy

35. 9.5 Rotational Work and Energy DEFINITION OF ROTATIONAL KINETIC ENERGY The rotational kinetic energy of a rigid rotating object is Requirement: The angular speed must be expressed in rad/s. SI Unit of Rotational Kinetic Energy: joule (J)

36. 9.5 Rotational Work and Energy Example 13 Rolling Cylinders A thin-walled hollow cylinder (mass = mh, radius = rh) and a solid cylinder (mass = ms, radius = rs) start from rest at the top of an incline. Determine which cylinder has the greatest translational speed upon reaching the bottom.

37. 9.5 Rotational Work and Energy ENERGY CONSERVATION

38. 9.5 Rotational Work and Energy The cylinder with the smaller moment of inertia will have a greater final translational speed.

39. 9.6 Angular Momentum DEFINITION OF ANGULAR MOMENTUM The angular momentum L of a body rotating about a fixed axis is the product of the body’s moment of inertia and its angular velocity with respect to that axis: Requirement: The angular speed must be expressed in rad/s. SI Unit of Angular Momentum: kg·m2/s

40. 9.6 Angular Momentum PRINCIPLE OF CONSERVATION OFANGULAR MOMENTUM The angular momentum of a system remains constant (is conserved) if the net external torque acting on the system is zero.

41. 9.6 Angular Momentum Conceptual Example 14 A Spinning Skater An ice skater is spinning with both arms and a leg outstretched. She pulls her arms and leg inward and her spinning motion changes dramatically. Use the principle of conservation of angular momentum to explain how and why her spinning motion changes.