Chapter 10 Rotational Motion 9-8 Center of Mass (CM) 9-9 Center of Mass and Translational Motion 10-1 Angular Quantities 10-2 Vector Nature of Angular Quantities 10-3 Constant Angular Acceleration HW#8: Due Friday, Nov 14 Chap. 9:Pb.10, Pb.25, Pb.35,Pb.50,Pb.56,Pb.64 Look at the example 9-11:Ballistic Pendulum Chapter Opener. Caption: You too can experience rapid rotation—if your stomach can take the high angular velocity and centripetal acceleration of some of the faster amusement park rides. If not, try the slower merry-go-round or Ferris wheel. Rotating carnival rides have rotational kinetic energy as well as angular momentum. Angular acceleration is produced by a net torque, and rotating objects have rotational kinetic energy.

9-8 Center of Mass (CM) In (a), the diver’s motion is pure translation; in (b) it is translation plus rotation. There is one point that moves in the same path a particle would take if subjected to the same force as the diver. This point is called the center of mass (CM). Figure 9-21. Caption: The motion of the diver is pure translation in (a), but is translation plus rotation in (b). The black dot represents the diver’s CM at each moment.

9-8 Center of Mass (CM) The general motion of an object can be considered as the sum of the translational motion of the CM, plus rotational, vibrational, or other forms of motion about the CM. Figure 9-22. Caption: Translation plus rotation: a wrench moving over a horizontal surface. The CM, marked with a red cross, moves in a straight line.

9-8 Center of Mass (CM) For two particles, the center of mass lies closer to the one with the most mass: where M is the total mass. Figure 9-23. Caption: The center of mass of a two-particle system lies on the line joining the two masses. Here mA > mB, so the CM is closer to mA than to mB .

9-8 Center of Mass (CM) Example 9-14: CM of three guys on a raft. Three people of roughly equal masses m on a lightweight (air-filled) banana boat sit along the x axis at positions xA = 1.0 m, xB = 5.0 m, and xC = 6.0 m, measured from the left-hand end. Find the position of the CM. Ignore the boat’s mass. Figure 9-24. Solution: For more than two particles, finding the center of mass is an extension of the two-particle process; just add the product of each mass and its distance from some reference point and divide by the total mass. This gives the distance of the CM from the reference point. Here it is 4.0 m.

9-8 Center of Mass (CM) Exercise 9-15: Three particles in 2-D. Three particles, each of mass 2.50 kg, are located at the corners of a right triangle whose sides are 2.00 m and 1.50 m long, as shown. Locate the center of mass. Figure 9-25. Solution: This is the same process as before, except that we have to do it separately for the x and y components of the position of the center of mass. We find x = 1.33 m and y = 0.50 m.

9-8 Center of Mass (CM) For an extended object, we imagine making it up of tiny particles, each of tiny mass, and adding up the product of each particle’s mass with its position and dividing by the total mass. In the limit that the particles become infinitely small, this gives: Figure 9-26. Caption: An extended object, here shown in only two dimensions, can be considered to be made up of many tiny particles (n), each having a mass Δmi. One such particle is shown located at a point ri = xii + yij + zik. We take the limit of n →∞ so Δmi becomes the infinitesimal dm.

9-8 Center of Mass (CM) Example 9-17: CM of L-shaped flat object. Determine the CM of the uniform thin L-shaped construction brace shown. Figure 9-29. Caption: This L-shaped object has thickness t (not shown on diagram). Solution: This object is basically two rectangles. Find the center of mass of each – they are shown on the diagram – then treat each as a point mass and find the overall center of mass. It is at x = 1.42 m, y = -0.25 m.

9-8 Center of Mass (CM) The center of gravity is the point at which the gravitational force can be considered to act. It is the same as the center of mass as long as the gravitational force does not vary among different parts of the object. Figure 9-30. Caption: Determining the CM of a flat uniform body.

9-8 Center of Mass (CM) The center of gravity can be found experimentally by suspending an object from different points. The CM need not be within the actual object—a doughnut’s CM is in the center of the hole. Figure 9-31. Caption: Finding the CG.

9-9 Center of Mass and Translational Motion The total momentum of a system of particles is equal to the product of the total mass and the velocity of the center of mass. The sum of all the forces acting on a system is equal to the total mass of the system multiplied by the acceleration of the center of mass: Therefore, the center of mass of a system of particles (or objects) with total mass M moves like a single particle of mass M acted upon by the same net external force.

10-1 Angular Quantities In purely rotational motion, all points on the object move in circles around the axis of rotation (“O”). The radius of the circle is R. All points on a straight line drawn through the axis move through the same angle in the same time. The angle θ in radians is defined: where l is the arc length. Figure 10-1. Caption: Looking at a wheel that is rotating counterclockwise about an axis through the wheel’s center at O (axis perpendicular to the page). Each point, such as point P, moves in a circular path; l is the distance P travels as the wheel rotates through the angle θ.

10-1 Angular Quantities Example 10-1: Birds of prey—in radians. A particular bird’s eye can just distinguish objects that subtend an angle no smaller than about 3 x 10-4 rad. (a) How many degrees is this? (b) How small an object can the bird just distinguish when flying at a height of 100 m? Figure 10-3. Caption: (a) Example 10–1. (b) For small angles, arc length and the chord length (straight line) are nearly equal. For an angle as large as 15°, the error in making this estimate is only 1%. For larger angles the error increases rapidly. Solution: a. 0.017° b. 3 cm (assuming the arc length and the chord length are the same)

10-1 Angular Quantities Angular displacement: The average angular velocity is defined as the total angular displacement divided by time: The instantaneous angular velocity: Figure 10-4. Caption: A wheel rotates from (a) initial position θ1 to (b) final position θ2. The angular displacement is Δθ = θ2 – θ1.

10-1 Angular Quantities The angular acceleration is the rate at which the angular velocity changes with time: The instantaneous acceleration:

10-1 Angular Quantities Every point on a rotating body has an angular velocity ω and a linear velocity v. They are related: Figure 10-5. Caption: A point P on a rotating wheel has a linear velocity v at any moment.

Two objects are sitting on a horizontal table that is undergoing uniform circular motion. Assuming the objects don’t slip, which of the following statements is true? A) Objects 1 and 2 have the same linear velocity, v, and the same angular velocity, . B) Objects 1 and 2 have the same linear velocity, v, and the different angular velocities, . C) Objects 1 and 2 have different linear velocities, v, and the same angular velocity, . D) Objects 1 and 2 have different linear velocities, v, and the different angular velocities, . Question 1 2