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Physics 2

Physics 2. Angular Momentum. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB. Angular Momentum.

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Physics 2

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  1. Physics 2 Angular Momentum Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  2. Angular Momentum This is the rotational equivalent of linear momentum. It quantifies the momentum of a rotating object, or system of objects. To get the angular momentum, we will start with linear momentum: This is the formula for linear momentum Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  3. Angular Momentum This is the rotational equivalent of linear momentum. It quantifies the momentum of a rotating object, or system of objects. To get the angular momentum, we will start with linear momentum: This is the formula for linear momentum Just like torque, we do a cross-product with the radius to get the angular quantity When the object is rotating about an axis of symmetry we can also use this formula Note that momentum is a vector quantity. Use your right-hand-rule for determining the direction. It will point the same direction as the angular velocity (perpendicular to r and p). Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  4. Angular Momentum Consider the simple case of a small mass (m) tied to a string with radius r. If the mass is swung around in a circle it will have some angular velocity . Notice that it will also have linear velocity (tangential to the circle). The relationship we know for these is v=r. m We can use this idea to find a useful alternate formula for angular momentum. The point mass will have a moment of inertia: r Substituting into the standard formula: We can use this formula when we have a point mass with a given linear velocity at some distance from a pivot point or axis of rotation. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  5. Angular Momentum We should have some formula relating angular momentum to torque (just like we have a formula relating linear momentum to force): This says that any time a torque is applied, there will be a corresponding change in angular momentum. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  6. Angular Momentum Like linear momentum, angular momentum is conserved. We will use this concept in several types of problems. This is a formula for conservation of angular momentum. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  7. Example: A 28 kg child is sitting on the edge of a 145 kg merry-go-round of radius 2.5 m while it is spinning at a rate of 3.7 rpm. If the child moves to the center, how fast will it be spinning? Assume the merry-go-round is a uniform cylinder. initial final r=2.5m Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  8. Example: A 28 kg child is sitting on the edge of a 145 kg merry-go-round of radius 2.5 m while it is spinning at a rate of 3.7 rpm. If the child moves to the center, how fast will it be spinning? Assume the merry-go-round is a uniform cylinder. We can use conservation of momentum for this one. Initially both the merry-go-round and the child contribute to the total angular momentum. However, once the child is at the center, she no longer has angular momentum (r=0). initial final r=2.5m Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  9. Example: A 28 kg child is sitting on the edge of a 145 kg merry-go-round of radius 2.5 m while it is spinning at a rate of 3.7 rpm. If the child moves to the center, how fast will it be spinning? Assume the merry-go-round is a uniform cylinder. We can use conservation of momentum for this one. Initially both the merry-go-round and the child contribute to the total angular momentum. However, once the child is at the center, she no longer has angular momentum (r=0). initial We need the total angular momentum of the system, Including both the child and the disk. final r=2.5m Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  10. Example: A 28 kg child is sitting on the edge of a 145 kg merry-go-round of radius 2.5 m while it is spinning at a rate of 3.7 rpm. If the child moves to the center, how fast will it be spinning? Assume the merry-go-round is a uniform cylinder. We can use conservation of momentum for this one. Initially both the merry-go-round and the child contribute to the total angular momentum. However, once the child is at the center, she no longer has angular momentum (r=0). initial We need the total angular momentum of the system, Including both the child and the disk. final r=2.5m The initial angular velocity is given in rpm, and we can leave it in those units for this type of problem. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  11. Example: A 28 kg child is sitting on the edge of a 145 kg merry-go-round of radius 2.5 m while it is spinning at a rate of 3.7 rpm. If the child moves to the center, how fast will it be spinning? Assume the merry-go-round is a uniform cylinder. We can use conservation of momentum for this one. Initially both the merry-go-round and the child contribute to the total angular momentum. However, once the child is at the center, she no longer has angular momentum (r=0). initial We need the total angular momentum of the system, Including both the child and the disk. final r=2.5m The initial angular velocity is given in rpm, and we can leave it in those units for this type of problem. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  12. Example: A helicopter rotor blade can be considered a long thin rod, as shown in figure below. • Each of the three rotor helicopter blades is L = 3.6 m long and has a mass of m = 150 kg. • While starting up, the motor provides a constant torque for 2 minutes, and the blades speed up from rest to 85 rad/s. • How much torque is produced by the motor? • How fast is the tip of each blade moving? Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  13. Example: A helicopter rotor blade can be considered a long thin rod, as shown in figure below. • Each of the three rotor helicopter blades is L = 3.6 m long and has a mass of m = 150 kg. • While starting up, the motor provides a constant torque for 2 minutes, and the blades speed up from rest to 85 rad/s. • How much torque is produced by the motor? • How fast is the tip of each blade moving? We will use angular momentum for this one. Since we have a constant torque, we simply need to find the change in angular momentum and divide by time. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  14. Example: A helicopter rotor blade can be considered a long thin rod, as shown in figure below. • Each of the three rotor helicopter blades is L = 3.6 m long and has a mass of m = 150 kg. • While starting up, the motor provides a constant torque for 2 minutes, and the blades speed up from rest to 85 rad/s. • How much torque is produced by the motor? • How fast is the tip of each blade moving? We will use angular momentum for this one. Since we have a constant torque, we simply need to find the change in angular momentum and divide by time. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  15. Example: A helicopter rotor blade can be considered a long thin rod, as shown in figure below. • Each of the three rotor helicopter blades is L = 3.6 m long and has a mass of m = 150 kg. • While starting up, the motor provides a constant torque for 2 minutes, and the blades speed up from rest to 85 rad/s. • How much torque is produced by the motor? • How fast is the tip of each blade moving? We will use angular momentum for this one. Since we have a constant torque, we simply need to find the change in angular momentum and divide by time. We need to find the moment of inertia for the blades. There are 3 of them, and each one is a long thin rod. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  16. Example: A helicopter rotor blade can be considered a long thin rod, as shown in figure below. • Each of the three rotor helicopter blades is L = 3.6 m long and has a mass of m = 150 kg. • While starting up, the motor provides a constant torque for 2 minutes, and the blades speed up from rest to 85 rad/s. • How much torque is produced by the motor? • How fast is the tip of each blade moving? We will use angular momentum for this one. Since we have a constant torque, we simply need to find the change in angular momentum and divide by time. We need to find the moment of inertia for the blades. There are 3 of them, and each one is a long thin rod. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  17. Example: A helicopter rotor blade can be considered a long thin rod, as shown in figure below. • Each of the three rotor helicopter blades is L = 3.6 m long and has a mass of m = 150 kg. • While starting up, the motor provides a constant torque for 2 minutes, and the blades speed up from rest to 85 rad/s. • How much torque is produced by the motor? • How fast is the tip of each blade moving? We will use angular momentum for this one. Since we have a constant torque, we simply need to find the change in angular momentum and divide by time. We need to find the moment of inertia for the blades. There are 3 of them, and each one is a long thin rod. Since the blades started from rest, this is ΔL. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  18. Example: A helicopter rotor blade can be considered a long thin rod, as shown in figure below. • Each of the three rotor helicopter blades is L = 3.6 m long and has a mass of m = 150 kg. • While starting up, the motor provides a constant torque for 2 minutes, and the blades speed up from rest to 85 rad/s. • How much torque is produced by the motor? • How fast is the tip of each blade moving? We will use angular momentum for this one. Since we have a constant torque, we simply need to find the change in angular momentum and divide by time. To find the speed of the blade tips, simply use v=r: We need to find the moment of inertia for the blades. There are 3 of them, and each one is a long thin rod. Since the blades started from rest, this is ΔL. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

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