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Angular momentum Tutorial homework due Thursday/Friday Conservation of angular momentum today Web page: 1 Conservation of angular momentum Previously found if
Angular momentum Tutorial homework due Thursday/Friday Conservation of angular momentum today Web page: 1 Conservation of angular momentum Previously found if no external force acts on a system, momentum is constant. We called this conservation of momentum. Similarly, if no external torque acts, angular momentum is constant. This is conservation of angular momentum.! Since τ net = d L!! d L!, if τ net = 0 then so is constant dt dt = 0! L Just like internal forces inside a system cannot produce a net force on the system due to Newton s 3 rd law, internal torques cannot produce a net torque so only external torques can change the total angular momentum of the system. Conservation of angular momentum is a powerful idea. For example, it explains a lot of astronomical observations. Conservation of angular momentum To solve problems in which momentum! is conserved we generally used the equation p f = p! i which we often write as: m! v f = m! v i Similarly, for angular momentum! conservation problems we will usually use the equation L f =!! L i, keeping in mind that L = r! p! = I ω!. Often we will use I f ω = I fz iω iz 3 Clicker question 1 Set frequency to BA A hollow sphere of mass M and radius 1.35 m is turning with an angular velocity of ω i = rad/s. This mass is compressed into a solid sphere of radius 0.45 m with no external torque applied. What can you say about the final angular velocity ω f? A. ω f ω i I I Isolid sphere = B. ω f = ω disk = 1 MR ring = MR 5 MR i C. ω f ω i I hollow sphere = 3 MR I rod (about center) = 1 1 ML D. Impossible to tell Conservation of angular momentum tells us: I f ω f = I i ω i I f I i because /5 /3 and R f R i Therefore ω f ω i. In fact, we can calculate ω f : ω f = I iω i I f = MR 3 i ω i = 5R i 3R ω i = f 5 MR f 5(1.35 m) 3(0.45 m) ω i = 15ω i = 30 rad/s 4 The spinning professor Consider a person with arms outstretched holding weights and the same person with arms tucked in holding weights. Could estimate moment of inertia for both situations using a cylinder for the body, rods for the arms, and point masses for the weights. I 1 I Clearly I 1 I so if there is an initial angular velocity ω 1 then since there is no external torque I 1 ω 1 = I ω so ω ω 1 Note if angular momentum is conserved but I 1 I and ω 1 ω 1 then rotational kinetic energy is not conserved: I 1ω 1 1 I ω Where does the work to change the kinetic energy come from? Force along a distance (work) is required to pull the weight in. 5 Clicker question Set frequency to BA A disk of mass M and radius R is rotating with angular velocity. Consider the effect on the angular velocity if a small object of mass m is dropped onto the disk. If the object is dropped at a radius of R the final angular velocity is ω 1 and if the object is dropped at a radius of R/ the final angular velocity is ω. What is the relationship between the various angular velocities? A. = ω 1 = ω B. C. D. E. ω ω 1 ω ω 1 ω 1 ω ω 1 ω Conservation of angular momentum gives us: ω 1 ω I 0 = I 1 ω 1 = I ω I 0 = 1 MR I 1 = 1 MR + mr I = 1 MR! + m # R I 1 I I0 so ω 0 ω ω1 6 $ & % More on clicker question Since there is no external torque, the overall angular momentum of the system remains the same which is how we solved the problem. ω0 ω1 ω Note that if we just consider the disk, the angular momentum decreased. Where did the torque come from? When the mass lands, friction acts to cause an acceleration of the mass and the equal and opposite force slows the disk. Need to carefully determine what is part of the system on which conservation of angular momentum can be applied. 7 New types of collision problems If an object comes in with some initial angular momentum then we need to account for that as well. p = mv ω = 0 ω 1 Initial angular momentum about the disk axis is L =! r! p = Rmv Angular momentum is conserved in the collision. The axis about which angular momentum is calculated must remain the same to use conservation of angular momentum! If the axis is fixed in place then momentum is not conserved (it is conserved when you consider the Earth as part of the system). If the axis is not fixed then momentum is also conserved. R 8 Clicker question 3 If I get on the rotating platform, initially at rest and start spinning a bicycle wheel clockwise whose axis points up, what will happen? A. I will start rotating clockwise B. I will start rotating counter-clockwise C. I will not rotate Set frequency to BA Initial angular momentum is 0 and no external torque is applied so final angular momentum must also be 0. Bicycle wheel has angular momentum pointing down (clockwise with vertical axis upward)! L i =! L W +! L P = = 0! L f =! L W +! L P = + = 0 Something else in the system must have angular momentum pointing in the opposite direction to have a total angular momentum of 0. ( Something else is a rotating professor.) What happens if I flip the bicycle wheel upside down? 9
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