Rotational Dynamics Lecture Slides presentation.pptx

University Physics

1. Chapt e r 9 Rotational D ynamics Cause of rotational motion

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 What causes an object to have an angular acceleration ? TORQUE, 𝑟 According to Newton’s second law, a net force causes an object to have an acceleration. 𝐚 = ∑ 𝐅 𝑚

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 D E FINI T ION OF T ORQUE Magnitude of Torque = (Magnitude of the force) x (Lever arm)   F 𝑙 Direction: The torque is positive when the force tends to produce a counterclockwise rotation about the axis. Lever ar m , 𝑃 is m ini m um distance between line of action and axis of rotation. SI Unit of Torque? Newton meter (N·m)

6. 9.1 The Action of Forces and Torques on Rigid Objects Example: T h e Achil l es T endon The tendon exerts a force of magnitude 790 N. Determine the torque (magnitude and direction) of this force about the ankle joint.   F 𝑙 𝑙  ( 3. 6  1  2 m ) cos 5 5 o 2.06 × 10 −2 𝑚   ( 79 N ) ( 2 .0 6  1  2 m ) 16.27 N.m

7. 9.2 Rigid Objects in Equilibrium If a rigid body is in equilibrium, neither its linear motion nor its rotational motion changes. Equilibrium  F x   F y     Linear motion a x  a y  Rotational motion  

8. 9.2 Rigid Objects in Equilibrium Write down what is known and what is unknown. Select the object to which the equations for equilibrium are to be applied. 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. (chose + and – directions) 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. (Chose counter clockwise + and clockwise -) Solve the equations for the desired unknown quantities. Reasoning Strategy

9. 9.2 Rigid Objects in Equilibrium Example: 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.    𝐹 2 𝑃 2 − 𝑤𝑃 𝑤 = 1476.4 N What if weight of board is 150 N?

10. 9.2 Rigid Objects in Equilibrium Example : Bodybuilding The arm is horizontal and weighs 31.0 N. The deltoid muscle can supply 1840 N of force. What is the weight of the heaviest dumbbell he can hold? 86.1 N

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

12. 9.3 Center of Gravity W 1  W 2    W 1 x 1  W 2 x 2   x cg When weight is not uniformly distributed

13. 9.3 Center of Gravity 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. Example: The Center of Gravity of an Arm The horizontal arm is composed of three parts: 0.278 m W 1  W 2  W 3  W 1 x 1  W 2 x 2  W 3 x 3 x cg

14. 9.3 Center of Gravity Conceptual Example: 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?

15. 9.3 Center of Gravity Finding the center of gravity of an irregular shape.

16. In rotational acceleration According to Newton’s second law, a net force causes an object to have an acceleration. ∑ 𝐅 = 𝐦𝐚    I           acceleration  M o m e nt of   A n g u l a r N e t e x t e rn a l t orque   i n e r t i a I    m r 2  I (moment of inertia) depends on how the mass is distributed around the axis of rotation. Units of I ? 𝑘𝑔 ∙ 𝑚 2 If an object is not in equilibrium For translation acceleration:

17. 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis Example: 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 one end and the center. 𝐼 = 𝑚 2 𝑟 2 2 𝐼 = 𝑚 1 𝑟 1 2 + 𝑚 2 𝑟 2 2 (a) (b) Table 9.1: Moment of inertia for various rigid bodies.

18. 9.5 Rotational Work and Energy W R   s  r  Work W  Fs W  Fr    Fr The rotational work is done by a constant torque in turning an object through an angle is SI Unit of Rotational Work? N.m = joule (J)

19. 9.5 Rotational Work and Energy 2 1 2 2 2  mr  1 2 2 2 2 1  m r      I  KE    2 2 T K E  1 m v 2  1 mr 2  2 v T  r  Kinetic Energy 1 2 𝐾𝐸 = 𝑚 𝑣 2 𝑅 2 1 𝐾𝐸 = 𝐼𝜔 2 SI Unit of Rotational Kinetic Energy? joule ( J )

20. 9.5 Rotational Work and Energy Mechanical energy Total mechanical Energy is equal to sum of translational kinetic energy, rotational kinetic energy, and gravitational energy E = KE T + KE R +PE E  1 m v 2  1 I  2  m g h 2 2

21. 9.6 Angular Momentum DEFINITION OF ANGULAR M O MENTUM 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: L  I  SI Unit of Angular Momentum? kg·m 2 /s 𝒑 = 𝑚𝒗

22. 9.6 Angular Momentum PRINCIPLE OF CONSERVATION OF ANGULAR MOMENTUM The angular momentum of a system remains constant (is conserved) if the net external torque acting on the system is zero.