PHYSICS 220 Lecture 15 Angular Momentum Textbook Sections 9.3 9.6 Lecture 15 Purdue University, Physics 220 1
Last Lecture Overview Torque = Force that causes rotation τ = F r sin θ Work done by torque W = τ θ Equilibrium Σ F = 0 Σ τ = 0 (Can choose any axis) Στ = Iα Rolling Object Two objects of the same mass rolling at the same translational speed do not necessarily have the same kinetic energy Today Angular Momentum L = Iω ΔL = 0 if Σ τ = 0 Lecture 15 Purdue University, Physics 220 2
Linear and Angular Linear Angular Displacement x θ Velocity v ω Acceleration a α Inertia m I Kinetic Energy ½ m v 2 ½ I ω 2 Newton s 2 nd Law F = ma τ = Iα Momentum p = mv L = Iω Lecture 15 Purdue University, Physics 220 3
Quiz 1) A uniform bar of mass m is supported by a pivot at its top, about which the bar can swing like a pendulum. If a force F is applied perpendicularly to the lower end of the bar as in the diagram, how big must F be in order to hold the bar in equilibrium at an angle from the vertical? A) 2mg B) mg sinθ C) 2mg sinθ D) (mg/2) sinθ E) (mg/2) cosθ Lecture 15 Purdue University, Physics 220 4
Quiz 2) You decide to roll two objects down a ramp to see which one gets to the bottom first. Object A is a solid cylinder and object B is a hollow cylinder. A and B have the same mass and radius. Suppose you released them from rest at the top of the ramp at same time, which has the most kinetic energy when it gets to the bottom of the ramp? A) Solid cylinder B) Hollow cylinder I solid = 1/2MR 2 C) Same I hollow = MR 2 Lecture 15 Purdue University, Physics 220 5
Define Angular Momentum Momentum Angular Momentum p = mv L = Iωω conserved if ΣF ext = 0 conserved if Στ ext = 0 Vector Vector units: kg-m/s units: kg-m 2 /s For a particle: L= Iω = mr = mvr = pr r Lecture 15 Purdue University, Physics 220 6 2 v
Right Hand Rule In general, for an object rotating about a fixed (z) axis we can write L z =I Iω The direction of L z is + CCW, - CW Wrap fingers on right hand, around angle of rotation, thumb points in direction of L L z =Iω Lecture 15 Purdue University, Physics 220 7 ω
Spinning Disks A disk of mass M and radius R rotates around the z axis with angular velocity ω i. A second identical disk, initially not rotating, is dropped on top of the first. There is friction between the disks, and eventually they rotate together with angular velocity ω f. A) ω f = ω i B) ω f = ½ ω i C) ω f = ¼ ω i z z ω i ω f Lecture 15 Purdue University, Physics 220 8
Two Spinning Disks First realize that there are no external torques acting on the two-disk system. Angular momentum will be conserved! L i = I + 0= 1 2 2 2 = = 1ω1+ MR ω L i f I1ω1+ I2ω2 MR ω f z 2 ω 0 1 1 MR ω = MR 2 2 2 i 1 ω = ω 2 i f ω z ω f Lecture 15 Purdue University, Physics 220 9
Demo: Spin Up/Down You are sitting on a freely rotating bar-stool with your arms stretched out and a heavy glass mug in each hand. Your friend gives you a twist and you start rotating around a vertical axis though the center of the stool. You can assume that the bearing the stool turns on is frictionless, and that there is no net external torque present once you have started spinning. You now pull your arms and hands (and mugs) close to your body. Lecture 15 Purdue University, Physics 220 10
Demo There are no external forces acting on the student+stool system. A) True B) False C) What!? FBD has gravity and normal force. Key is no external torques! Lecture 15 Purdue University, Physics 220 11
Demo: Spin Up/Down What happens to the angular momentum as you pull in your arms? A) it increases B) it decreases C) it stays the same torques no external forces gives constant angular momentum L 1 L 2 Lecture 15 Purdue University, Physics 220 12
Demo: Spin Up/Down What happens to your angular velocity as you pull in your arms? A) it increases B) it decreases ω ω 1 C) it stays the same 2 I 1 I 2 L L Angular velocity increases because inertia decreases Pulling in your arms decreases your moment of inertia. Therefore, you will have an increased angular velocity Lecture 15 Purdue University, Physics 220 13
Demo: Spin Up/Down What happens to your kinetic energy as you pull in your arms? A) it increases B) it decreases C) it stays the same ω 1 ω 2 KE = 1 I 2 ω = 2 1 I 2 2 ω 2I = 1 2I L 2 (using L = Iω ) I 1 I 2 L L The mass is closer so you have less inertia and you speed up. If you speed up then, your kinetic energy must increase as well. Lecture 15 Purdue University, Physics 220 14
Sliding Puck A puck slides in a circular path on a horizontal frictionless table. It is held at a constant radius by a string threaded d through a frictionless hole at the center of the table. If you pull on the string such that the radius decreases by a factor of 2, by what factor does the angular velocity of the puck increase? A) 2 B) 4 C) 8 ω Lecture 15 Purdue University, Physics 220 15
Solution Since the string is pulled through a hole at the center of rotation, ti there is no torque: Angular momentum is conserved. L 1 = I 1 ω 1 = mr 2 ω 1 = R 2 L 2 = I 2 ω 2 = m ω 2 2 mr 2 ω = 2 1 1/4 mr ω 2 ω 1 =1/4 ω 2 ω 2 = 4ω 1 m R ω 1 m R/2 ω 2 Lecture 15 Purdue University, Physics 220 16
Turning the Bike Wheel A student sits on a barstool holding a bike wheel. The wheel is initially spinning CCW in the horizontal plane (as viewed from above) L= 25 kg m 2 /s. She now turns the bike wheel over. What happens? A) She starts to spin CCW B) She starts to spin CW C) Nothing Start with angular momentum L pointing up from wheel. When wheel is flipped, no more angular momentum from it pointing up, so need to spin person/stool to conserve L! Lecture 15 Purdue University, Physics 220 17
Turning the Bike Wheel She is holding the bike wheel and spinning counter clockwise. What happens if she turns it the other ½ rotation (so it is basically upside down from how it started). A) Spins Faster B) Stays same C) Stops Lecture 15 Purdue University, Physics 220 18
Turning the Bike Wheel Since there is no net external torque acting on the studentstool system, angular momentum is conserved. Remenber, L has a direction as well as a magnitude! Initially: L INI = L WI W,I = + 25 kg m 2 /s Finally: L FIN = L W,F + L S = -25 kg m 2 /s +L s L s = 50 kg m 2 /s L S L W,I L W,F L W,I = L W,F + L S Lecture 15 Purdue University, Physics 220 19
Gyroscopic Motion Suppose you have a spinning gyroscope in the configuration shown below: If the left support is removed, what will happen? support g ω pivot Lecture 15 Purdue University, Physics 220 20
Gyroscopic Motion Suppose you have a spinning gyroscope in the configuration shown below: If the left support is removed, what will happen? The gyroscope does not fall down! ω pivot g Lecture 15 Purdue University, Physics 220 21
Gyroscopic Motion... instead it precesses around its pivot axis! ω pivot Lecture 15 Purdue University, Physics 220 22
Gyroscope Because of the way the wheel is mounted in the frame, the torque on the wheel is zero Even when the frame is moved or rotated The wheel s angular momentum is conserved The orientation of the gyroscope provides a direction finder Lecture 15 Purdue University, Physics 220 23
Earth as a Gyroscope The Earth acts as a gyroscope as it spins on its axis This is a spin angular momentum, separate from its orbital angular momentum The spin angular momentum points in the direction of the north pole There e is no external torque on the Earth so its spin angular momentum is conserved The Earth s spin axis is tilted about 23.5 from the perpendicular to the orbital axis Since its spin angular momentum is conserved, the rotational axis remains tilted at a fixed angle with respect to the orbital plane This produces seasons on the Earth Lecture 15 Purdue University, Physics 220 24
Spinning Wheel A spinning wheel is more stable than a stationary one The increased stability is due to its angular momentum The angular momentum is directed along the axis of the wheel and so, ideally, there is no external torque on the wheel and it would remain in the same direction and never fall over However, in reality there is some small external torque and so the wheel will eventually tip over This torque could be from friction or deviations in the shape of the wheel, etc. Lecture 15 Purdue University, Physics 220 25
Precession The external torque on a rotating object can be substantial This leads to an effect called precession At rest, the device s stability is very low Its angular momentum adds to its stability, although the gravitational forceactingonitislarge on it is Assume the gravitational force acts on the center of mass of the system The center of mass of the system is close to the center of the wheel The applied torque leads to a change in the angular momentum of the system τ Δ t =Δ L Lecture 15 Purdue University, Physics 220 26
Example: Bullet Hitting Stick A uniform stick of mass M and length D is pivoted at the center. A bullet of mass m is shot through the stick at a point halfway between the pivot and the end. The initial speed of the bullet is v 1, and the final speed is v 2. What is the angular speed ω F of the stick after the collision? (Ignore friction and gravity) M m D D/4 ω F v 1 v 2 initial final Lecture 15 Purdue University, Physics 220 27
Example: Bullet Hitting Stick Conserve angular momentum around the pivot (z) axis! The total angular momentum before the collision is due only to the bullet (since the stick is not rotating gy yet). M m D D/4 v 1 initial Lecture 15 Purdue University, Physics 220 28
Example: Bullet Hitting Stick Conserve angular momentum around the pivot (z) axis! The total angular momentum after the collision has contributions from both the bullet and the stick where I is the moment of inertia of the stick about the pivot. D/4 ω F v 2 final Lecture 15 Purdue University, Physics 220 29
Example: Bullet Hitting Stick Set L i = L f : mv 1 (D/4) = mv 2 (D/4) + Iω F m M D D/4 ω F v 1 v 2 initial final Lecture 15 Purdue University, Physics 220 30
Example: Bullet Hitting Stick Set L i = L f using m M D D/4 ω F v 1 v 2 initial final Lecture 15 Purdue University, Physics 220 31
Στ = I α Summary of Concepts Energy is Conserved Need to include translational and rotational L = I ω Right Hand Rule gives direction If Στ = 0, L is conserved Lecture 15 Purdue University, Physics 220 32