Physics 131: Lecture Today s Agenda Rotational dynamics Torque = I Angular Momentum Physics 01: Lecture 10, Pg 1
An Unfair Race A frictionless block and a rolling (without slipping) disk are released at the top of identical inclined planes. Which will reach the bottom first? (a) (b) (c) Frictionless block Rolling Disk The same time H H Physics 01: Lecture 10, Pg
Clicker Question 1: A frictionless block and a rolling (without slipping) disk are released at the top of identical inclined planes. Which will reach the bottom first? (A) The disk loses energy to friction as it rolls, but the box is frictionless and so it speeds up more quickly and gets to the bottom first. (B) The potential energy of the disk is converted into translational and rotational kinetic energy, so the translational speed grows more slowly than that of the box, which has no rotational energy. (C)The net forces on the two objects are equal, but the force on the disk gets partially used up in creating the torque necessary to make it roll. (D) The net forces on the two objects are equal, but the force on the disk is not directed parallel to the ramp, and so does not create as great an acceleration down the ramp. Physics 01: Lecture 10, Pg 3
An Unfair Race A frictionless block and a rolling (without slipping) disk are released at the top of identical inclined planes. Which will reach the bottom first? KTOT 1 MV Doesn t friction take away energy? H K TOT 1 MV 1 I H There is only one source of energy, gravitational potential energy. For the disk that is rotating it must spend this energy to rotate and to move down the incline, so it will have less energy to move down the incline and thus will move slower. Physics 01: Lecture 10, Pg 4
An Unfair Race A frictionless block and a rolling (without slipping) disk are released at the top of identical inclined planes. Which will reach the bottom first? H H Physics 01: Lecture 10, Pg 5
Displacement s Linear and Angular Linear Angular Velocity v Acceleration a Inertia m I K ½ m v ½ I NL F = ma = I Momentum p = mv L = Iw Today Physics 01: Lecture 10, Pg 6
Angular momentum of a rigid body about a fixed axis: Here we have a disk with moment of inertia and angular velocity It has angular momentum of Rotation is positive (CounterClockwise) z L I Angular momentum is a vector! Physics 01: Lecture 10, Pg 7
The magnitude of the angular velocity vector is. The angular velocity vector points along the axis of rotation in the direction given by the right-hand rule as illustrated. Angular Momentum Physics 01: Lecture 10, Pg 8
z External/Internal Torques i f d L net dt d L ext int net net dt d L ext 0 net dt d L ext net dt Physics 01: Lecture 10, Pg 9
Conservation of Angular Momentum Newton s second law for rotations ext dl dtt When the external torques on an system sum to zero, the total angular momentum of the system will be conserved The angular momentum of an isolated system is conserved Physics 01: Lecture 10, Pg 10
Conservation of Linear Momentum The angular momentum of an isolated system is conserved L L f i I I f i object 1 object object 1 object I f + I f = I i + I i Physics 01: Lecture 10, Pg 11
Demonstrations: Angular momentum What happens to your angular momentum as you pull in your arms? Does it increase, decrease or stay the same? What happens to your angular velocity as you pull in your arms? i f I i I f Physics 01: Lecture 10, Pg 1
Example: Rotating Table... There are no external torques acting on the student-stool system, so angular momentum will be conserved. Initially: L i = I i i Finally: L f = I f f f I i 1 I i I f i f I i I f Physics 01: Lecture 10, Pg 13
Clicker Question : What happens to your kinetic energy as you pull in your arms? (a) (b) (c) increases decreases stays the same No external torques, ang mom is conserved i i f f I i I i I f I f Physics 01: Lecture 10, Pg 14
Clicker Question 3: 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= 5 kg m /s She now turns the bike wheel over. What happens? (a) She starts to spin CCW. (b) She starts to spin CW. (c) Nothing Physics 01: Lecture 10, Pg 15
Clicker Question 4: Suppose you are standing on the center of a merry-go-round round that is at rest. You are holding a spinning bicycle wheel over your head so that its rotation axis is pointing upward. The wheel is rotating counterclockwise when observed from above. (For this problem, neglect any air resistance or friction between the merry-go-round and its foundation.) Suppose you now grab the edge of the wheel with your hand, stopping it from spinning. What happens to the merry-goround? (A) It remains at rest. (B) It begins to rotate CCW (as observed from above). (C) It begins to rotate CW (as observed from above). Physics 01: Lecture 10, Pg 16
Rotating Neutron Star A star is dying. Its core collapses such that its radius goes from 6,000 km to about 40 km. Assume the core does not lose mass. If it were rotating at an angular speed of 0.1 rad/s, what is its speed now? How many times per second does this revolve? Physics 01: Lecture 10, Pg 17
Clicker Question 5: A star is dying. Its core collapses such that its radius goes from 6, 000 km to about 40 km. If it was rotating at an angular speed of 0.1 rad/s, what is its speed now? Assume no outside torques act and the core does not lose mass. (a) 700 rad/s (b) 4600 rad/s (c) 436 rad/s (d) 18 rad/s (e) 0.1 rad/s Physics 01: Lecture 10, Pg 18
Clicker Question 6: Two astronauts a distance d I apart of mass M are spinning with an angular velocity of around an axis about their center. They pull on the rope so they are now a distance of d F apart. What is the initial angular momentum of the astronauts? (a) M (d I ) (b) M (d I ) (c) ½ M (d I ) (d) ¼ M (d I ) (e) 4 M (d I ) M d I M Physics 01: Lecture 10, Pg 19
Clicker Question 6: Two astronauts 35 m apart of mass 75 kg are spinning with an angular velocity of 45 rad/s around an axis about their center. They pull on the rope so they are now a distance of 15 m apart. What angular speed do they rotate with now? (a) 345 rad/s (b) 300 rad/s (c) 45 rad/s (d) 45 rad/s (e) 90 rad/s Physics 01: Lecture 10, Pg 0
Clicker Question 7: F L I L 1 1 1 1 I F F Md Md I F F d d I F d d F d Physics 01: Lecture 10, Pg 1
Clicker Question 8: You want to measure the moment of inertia for a rotating oval object. Initially, this object has an angular speed of 35 rad/s. You drop a ring of mass 30 kg and radius 0. m on top of the object. The ring and object apply frictional forces on each other and eventually come to have the same angular speed of 4 rad/s. What is the moment of inertia of the object? ring (a) 4.67 kg m (b).6 kg m (c) 3.45 kg m (d) 8.9 kg m (e) 1.3 kg m i f initial No external torques, ang mom is conserved final Physics 01: Lecture 10, Pg
Clicker Question 8: We can find the angular momentum: And then set them equal: L L I mr f I i o i f o f I o i I o mr f I o mr f i f.6 kg m initial i final f Physics 01: Lecture 10, Pg 3