Physics 6A. Angular Momentum. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
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1 Phyic 6A Angular Momentum For Campu earning
2 Angular Momentum Thi i the rotational equivalent of linear momentum. t quantifie the momentum of a rotating object, or ytem of object. f we imply tranlate the variable, we can get one poible equation: p m v Thi i the formula for linear momentum For Campu earning
3 Angular Momentum Thi i the rotational equivalent of linear momentum. t quantifie the momentum of a rotating object, or ytem of object. f we imply tranlate the variable, we can get one poible equation: p m v Thi i the formula for linear momentum Uing intead of m and intead of v, we get thi formula for angular momentum Note that momentum i a vector quantity. We probably will not need to conider that apect of it in thi cla, however. f you are curiou about what you are miing, check out the ection in your book about gyrocope. Good tuff. For Campu earning
4 Angular Momentum Conider the imple cae of a mall ma (m tied to a tring with radiu r. f the ma i wung around in a circle it will have ome angular velocity. Notice that it will alo have linear velocity (tangential to the circle. The relationhip we know for thee i v=r. m We can ue thi idea to find a ueful alternate formula for angular momentum. r The point ma will have a moment of inertia: m r Subtituting into the tandard formula: (m pointm a r v ( r m v r We can ue thi formula when we have a point ma with a given linear velocity at ome ditance from a pivot point or axi of rotation. For Campu earning
5 Angular Momentum ike linear momentum, angular momentum i conerved. We will ue thi concept in everal type of problem. f f i i Thi i a formula for conervation of angular momentum. For Campu earning
6 Angular Momentum ike linear momentum, angular momentum i conerved. We will ue thi concept in everal type of problem. f f i i Thi i a formula for conervation of angular momentum. Alo, we hould have ome formula relating angular momentum to torque (jut like we have a formula relating linear momentum to force: t Thi ay that any time a torque i applied, there will be a correponding change in angular momentum. Again, we probably won t need the vector apect here, but we might need +/- ign to keep track of it. For Campu earning
7 Example: A 8 kg child i itting on the edge of a 45 kg merry-go-round of radiu.5 m while it i pinning at a rate of 3.7 rpm. f the child move to the center, how fat will it be pinning? Aume the merry-go-round i a uniform cylinder. initial final r=.5m For Campu earning
8 Example: A 8 kg child i itting on the edge of a 45 kg merry-go-round of radiu.5 m while it i pinning at a rate of 3.7 rpm. f the child move to the center, how fat will it be pinning? Aume the merry-go-round i a uniform cylinder. We can ue conervation of momentum for thi one. nitially both the merry-go-round and the child contribute to the total angular momentum. However, once the child i at the center, he no longer ha angular momentum (r=0. initial final r=.5m For Campu earning
9 Example: A 8 kg child i itting on the edge of a 45 kg merry-go-round of radiu.5 m while it i pinning at a rate of 3.7 rpm. f the child move to the center, how fat will it be pinning? Aume the merry-go-round i a uniform cylinder. We can ue conervation of momentum for thi one. nitially both the merry-go-round and the child contribute to the total angular momentum. However, once the child i at the center, he no longer ha angular momentum (r=0. f i We need the total angular momentum of the ytem, ncluding both the child and the dik. initial dik,f dik,f dik,i dik,i girl,i girl,i final r=.5m For Campu earning
10 Example: A 8 kg child i itting on the edge of a 45 kg merry-go-round of radiu.5 m while it i pinning at a rate of 3.7 rpm. f the child move to the center, how fat will it be pinning? Aume the merry-go-round i a uniform cylinder. We can ue conervation of momentum for thi one. nitially both the merry-go-round and the child contribute to the total angular momentum. However, once the child i at the center, he no longer ha angular momentum (r=0. f i We need the total angular momentum of the ytem, ncluding both the child and the dik. initial dik,f dik,f dik,i dik,i girl,i girl,i final r=.5m The initial angular velocity i given in rpm, and we can leave it in thoe unit for thi type of problem. For Campu earning
11 Example: A 8 kg child i itting on the edge of a 45 kg merry-go-round of radiu.5 m while it i pinning at a rate of 3.7 rpm. f the child move to the center, how fat will it be pinning? Aume the merry-go-round i a uniform cylinder. We can ue conervation of momentum for thi one. nitially both the merry-go-round and the child contribute to the total angular momentum. However, once the child i at the center, he no longer ha angular momentum (r=0. f i We need the total angular momentum of the ytem, ncluding both the child and the dik. initial dik,f dik,f dik,i dik,i girl,i girl,i final r=.5m The initial angular velocity i given in rpm, and we can leave it in thoe unit for thi type of problem. ( m r dik dik dik,f ( m r dik dik ( i (m r girl girl ( i ( 45kg dik,f (.5m 5.rpm dik,f ( 45kg (.5m (3.7rpm (8kg (.5m (3.7rpm For Campu earning
12 Example: A helicopter rotor blade can be conidered a long thin rod, a hown in figure below. Each of the three rotor helicopter blade i = 3.6 m long and ha a ma of m = 50 kg. While tarting up, the motor provide a contant torque for minute, and the blade peed up from ret to 85 rad/. How much torque i produced by the motor? How fat i the tip of each blade moving? For Campu earning
13 Example: A helicopter rotor blade can be conidered a long thin rod, a hown in figure below. Each of the three rotor helicopter blade i = 3.6 m long and ha a ma of m = 50 kg. While tarting up, the motor provide a contant torque for minute, and the blade peed up from ret to 85 rad/. How much torque i produced by the motor? How fat i the tip of each blade moving? We will ue angular momentum for thi one. Since we have a contant torque, we imply need to find the change in angular momentum and divide by time. t For Campu earning
14 Example: A helicopter rotor blade can be conidered a long thin rod, a hown in figure below. Each of the three rotor helicopter blade i = 3.6 m long and ha a ma of m = 50 kg. While tarting up, the motor provide a contant torque for minute, and the blade peed up from ret to 85 rad/. How much torque i produced by the motor? How fat i the tip of each blade moving? We will ue angular momentum for thi one. Since we have a contant torque, we imply need to find the change in angular momentum and divide by time. t Ue the definition of angular momentum: For Campu earning
15 Example: A helicopter rotor blade can be conidered a long thin rod, a hown in figure below. Each of the three rotor helicopter blade i = 3.6 m long and ha a ma of m = 50 kg. While tarting up, the motor provide a contant torque for minute, and the blade peed up from ret to 85 rad/. How much torque i produced by the motor? How fat i the tip of each blade moving? We will ue angular momentum for thi one. Since we have a contant torque, we imply need to find the change in angular momentum and divide by time. t Ue the definition of angular momentum: We need to find the moment of inertia for the blade. There are 3 of them, and each one i a long thin rod. rod M 3 For Campu earning
16 Example: A helicopter rotor blade can be conidered a long thin rod, a hown in figure below. Each of the three rotor helicopter blade i = 3.6 m long and ha a ma of m = 50 kg. While tarting up, the motor provide a contant torque for minute, and the blade peed up from ret to 85 rad/. How much torque i produced by the motor? How fat i the tip of each blade moving? We will ue angular momentum for thi one. Since we have a contant torque, we imply need to find the change in angular momentum and divide by time. t Ue the definition of angular momentum: We need to find the moment of inertia for the blade. There are 3 of them, and each one i a long thin rod. rod M 3 3( 3 M (50kg(3.6m (85 rad 65,40 kgm For Campu earning
17 Example: A helicopter rotor blade can be conidered a long thin rod, a hown in figure below. Each of the three rotor helicopter blade i = 3.6 m long and ha a ma of m = 50 kg. While tarting up, the motor provide a contant torque for minute, and the blade peed up from ret to 85 rad/. How much torque i produced by the motor? How fat i the tip of each blade moving? We will ue angular momentum for thi one. Since we have a contant torque, we imply need to find the change in angular momentum and divide by time. t Ue the definition of angular momentum: We need to find the moment of inertia for the blade. There are 3 of them, and each one i a long thin rod. rod M 3 3( 3 M (50kg(3.6m (85 rad 65,40 kgm Since the blade tarted from ret, thi i Δ. 65,40 0 kgm,377n m For Campu earning
18 Example: A helicopter rotor blade can be conidered a long thin rod, a hown in figure below. Each of the three rotor helicopter blade i = 3.6 m long and ha a ma of m = 50 kg. While tarting up, the motor provide a contant torque for minute, and the blade peed up from ret to 85 rad/. How much torque i produced by the motor? How fat i the tip of each blade moving? We will ue angular momentum for thi one. Since we have a contant torque, we imply need to find the change in angular momentum and divide by time. t Ue the definition of angular momentum: We need to find the moment of inertia for the blade. There are 3 of them, and each one i a long thin rod. rod M 3 To find the peed of the blade tip, imply ue v=r : v v (3.6m 306 m (85 rad 3( 3 M (50kg(3.6m (85 rad 65,40 kgm Since the blade tarted from ret, thi i Δ. 65,40 0 kgm,377n m For Campu earning
Physics 2. Angular Momentum. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Phyic Angular Momentum For Campu earning Angular Momentum Thi i the rotational equivalent of linear momentum. t quantifie the momentum of a rotating object, or ytem of object. To get the angular momentum,
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