Rotational Kinetic Energy Conservation of Angular Momentum Vector Nature of Angular Quantities

8.7 Rotational Kinetic Energy All particles making up a rotating rigid body rotate with the same angular velocity . This gives us a starting equation for rotational kinetic energy: The subscript “p” is there to remind you that the moment of inertia I depends on the pivot point “p,” and therefore so does the rotational KE. KE = (mv 2 /2) = (m(r) 2 /2) = (1/2) (mr) 2  2 = I 2. OSE: K p = ½ I p  p 2. The kinetic energy of the rotating body is

Example: a solid cylinder and a thin hollow cylinder, both of mass M and outer radius R, are released from rest at the start of an inclined plane of height H and length L. What are the speeds of each cylinder when they reach the bottom? Which cylinder gets to the bottom first? If an object is rolling (with axis of rotation through its center of mass) while undergoing translational motion, then OSE: K total = K trans + K rot = ½ MV CM 2 + ½ I CM  CM2. “Nearly-official” solution that I scanned for Winter/Spring 2002 is on next page.

Example: Physics 23 Problem on KE of Rotation (from Test 3 Winter 2001). A cylinder of mass M, radius R and length L can freely rotate about a light metal rod through its center. The ends of the rod are attached to a massless yoke that can be used to pull the cylinder along a surface. D P=2Mg r M  The cylinder is placed at rest on a horizontal surface. The yoke is then pulled by a worker with a constant force of magnitude P=2Mg at an angle of  with respect to the vertical. It rolls without slipping at all times.

d D P=2Mg r M  Derive an expression for the linear speed of the cylinder, in terms of relevant system parameters, after it has rolled a distance D. V i =0 Vf?Vf? For an OSE you can use E f -E i =[W other ] i → f or [W net ] i → f =K. Because the only force that does work is P, in either equation, the calculation of work is the same. There is no change of height and there are no springs, so the K equation is “easier.” K = PD cos(90-) = PD sin  K = W P = P·d → → 90-

D P=2Mg r M  V i =0 Vf?Vf? d K f - K i = PD sin  ½ M V cm,f 2 + ½ I  f 2 = PD sin  ½ M V f 2 + ½ (½ Mr 2 ) (V f /r) 2 = (2Mg)D sin  The cylinder is solid (if that is not clear, you should ask) and has a moment of inertia ½ Mr 2. ½ V f 2 + ¼V f 2 = 2gD sin  ¾V f 2 = 2gD sin  V f 2 = (8gD sin )/3 90- 0

8.8 Angular Momentum and its Conservation Recall that linear momentum is defined by p = mv. The rotational analog of m is I, and the rotational analog of v is . We define angular momentum of a rigid, rotating body by From the definition of torque, if  z = 0, then  z = 0,  z is constant, and L z is constant. OSE: L z = I z. Linear momentum p is the fundamental descriptor of translational motion. It is changed by forces. Angular momentum L is the fundamental descriptor of rotational motion. It is changed by torques.

The total angular momentum of a rotating body remains constant if the net torque acting on it is zero. If the net torque is zero, angular momentum is con- served. This is another of the fundamental conservation laws of physics. OSE:  z,ext = 0  L z,i = L z,f. z v M RR L z,particle  = R  Mv. I will use this in working example A particle moving in a straight line with velocity v has an angular momentum relative to an axis.

Example A mass M attached to the end of a string revolves in a circle on a frictionless tabletop. The other end of the string passes through a hole in the table. Initially the mass revolves with a speed V 1 in a circle or radius R 1. The string is then pulled through the hole, so that the radius is reduced to R 2. Calculate the speed of the mass now. You can use the equation on the previous slide for the angular momentum of a particle, or you can use the moment of inertia of a single particle MR 2 from section 8.5, or you can treat the rotating particle like a thin hoop of mass M and radius R.

8.9 Vector Nature of Angular Quantities I have already introduced you to the ideas in this section. You will not be tested on it. If there is time, I will do demonstrations. Demonstration: spinning professor makes himself sick. Demonstration: bicycle wheel gyroscope. The following demonstration introduces the next chapter. Demonstration: sacrificial broom.