PHYSICS  CLUTCH CH 14: ANGULAR MOMENTUM.


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2 EXAMPLE: HOLDING WEIGHTS ON A SPINNING STOOL EXAMPLE: You stand on a stool that is free to rotate about an axis perpendicular to itself and through its center. Suppose that your combined moment of inertia (you + stool) around this axis is 8 kg m 2, and that it is the same whether you have your arms stretched open, or pressed against your body (not true, but simpler). You stand on the stool with arms wide open, holding one 10 kg small weight on each hand, at a distance of 80 cm from the central axis. (a) Calculate the system s total (you + stool + 2 weights) moment of inertia about its central axis. (b) Suppose the system spins with 60 RPM. Calculate its angular moment about its central axis. (c) You bring the weights to your chest, so they lie on the axis of rotation. Calculate the system s new moment of inertia. (d) You bring the weights to your chest, so they lie on the axis of rotation. How fast, in RPM, will the system rotate now? Page 2
3 PRACTICE: CLOSING YOUR ARMS ON A SPINNING STOOL PRACTICE: You stand on a stool that is free to rotate about an axis perpendicular to itself and through its center. The stool s moment of inertia around its central axis is 1.50 kg m 2. Suppose you can model your body as a vertical solid cylinder (height = 1.80 m, radius = 20 cm, mass = 80 kg) with two horizontal thin rods as your arms (each: length = 80 cm, mass = 3 kg) that rotate at their ends, about the same axis, as shown. Suppose that your arms contribution to the total moment of inertia is negligible if you have them pressed against your body, but significant if you have them wide open. If you initially spin at 5 rad/s with your arms against your body, how fast will you spin once you stretch them wide open? (Note: The system has 4 objects (stool + body + 2 arms), but initially only stool + body contribute to its moment of inertia) Page 3
4 CONSERVATION OF ANGULAR MOMENTUM Remember: The most important part about LINEAR Momentum was that it is.  The same is true for ANGULAR Momentum. Most Angular Momentum problems are about. LINEAR MOMENTUM ( p = ) ANGULAR MOMENTUM ( L = )  Conserved if no external.  Or better yet, if Σ.  MOST problems involve TWO objects.  Conservation:.  Conserved if no external.  Or better yet, if Σ.  MANY problems involve just ONE object.  Conservation:.  TWO Objects: PushAway; Collision  Add/remove mass (collision/pushaway)  ONE Object: Objects:. EXAMPLE: For each of the following, indicate whether the angular speed will increase or decrease: (a) An ice skater spins on frictionless ice. What happens to her angular speed if she closes her arm? (b) A large horizontal disc spins around itself. What happens to the disc s angular speed if you land on it? (c) An object tied to a point via a string spins horizontally around it. What happens if you shorten the string? (d) A star (like the Sun) spins around itself. What happens if it collapses and loses half its mass and half its radius? Page 4
5 EXAMPLE: ICE SKATER CLOSES HER ARMS EXAMPLE: Suppose an ice skater has moment of inertia of 6 kg m 2 while spinning with arms wide open, and 4 kg m 2 if she closes her arms. If she spins at 120 RPM with arms open, what RPM will she have as a result of closing her arms? Page 5
6 PRACTICE: DIVER CHANGING ANGULAR SPEED PRACTICE: Suppose a diver spins at 8 rad/s while falling with a moment of inertia about an axis through himself of 3 kg m 2. What moment of inertia would the diver need to have to spin at 4 rad/s? BONUS: How could he accomplish this? Page 6
7 EXAMPLE: ANGULAR SPEED OF STAR AFTER COLLAPSE EXAMPLE: When a star exhausts all its stellar energy, it dies, at which point a gravitational collapse happens, causing its radius and mass to decrease substantially. Our Sun spins around itself, at its equator, every 24.5 days. If our Sun were to collapse and shrink 90% in mass and 90% in radius, how long would its new period of rotation take, in days? Page 7
8 PRACTICE: TWO ASTRONAUTS SPINNING IN SPACE PRACTICE: Two astronauts, both 80 kg, are connected in space by a light cable. When they are 10 m apart, they spin about their center of mass with 6 rad/s. Calculate the new angular speed they ll have if they pull on the rope to reduce their distance to 5 m. You may treat them as point masses, and assume they continue to spin around their center of mass. Page 8
9 EXAMPLE: MOVING ON A ROTATING DISC EXAMPLE: A 200 kg disc 4 m in radius spins around a perpendicular axis through its center at 2 rad/s. An 80 kg person falls into the disc, with no horizontal speed, at a point 3 m away from the disc s center. Treating the person as a point mass: (a) Calculate the disc s new angular speed. (b) Calculate the person s new tangential speed. (c) The person starts walking towards the disc s center. Calculate the disc s speed once the person reaches it. Page 9
10 ANGULAR MOMENTUM AND NEWTON S SECOND LAW Remember: Newton s Second Law,, can be rewritten in terms of Linear Momentum:  The Rotational version,, can also be rewritten, in terms of Angular Momentum: EXAMPLE: A small object, 10 kg in mass, spins at 180 RPM in a circular path of radius 5 m. If a constant torque of 80 N m is applied to the object, in trying to stop it, how long will it take for the object to come to a complete stop? Page 10
11 PRACTICE: ANGULAR MOMENTUM / FORCE TO ACCELERATE PRACTICE: A solid disc of mass M = 40 kg and radius R = 2 m is free to rotate about a fixed, frictionless, perpendicular axis through its center. You apply a constant, tangential force on the disc s surface (as shown), to get it to spin. Calculate the magnitude of the force needed to get the disc to 100 rad/s in just one minute. Page 11
12 ANGULAR COLLISIONS Angular Collisions happen when ONE of the TWO objects involved in the collision is rotating OR rotates as a result.  Similar to Linear Collisions: We use the Conversation of Momentum equation, but the version: = =  For a Point Mass in linear motion, we use the linear version of the Angular Momentum equation L =.  Where is the distance between where the linear object collides and the axis of the rotating object.  Note: Adding mass to a rotating disc is an Angular Collision problem; but simple if the mass was at rest. EXAMPLE: A 100 kg solid disc of radius 6 m spins clockwise around a perpendicular axis through its center at 120 RPM. A second solid disc, 50 kg in mass, 3 m in radius, is carefully placed on top of the first disc, as shown, causing the discs to spin together about the same axis. Calculate the new rate, in RPM, that the discs will have, if the second disc was initially: (a) at rest; (b) spinning counterclockwise with 360 RPM. Page 12
13 EXAMPLE: JUMPING INTO A MOVING DISC EXAMPLE: A 200 kg disc 4 m in radius spins around a perpendicular axis through its center at 2 rad/s counterclockwise. Find the new speed the disc would have if an 80 kg person (treat as a point mass): (a) Steps into the disc s edge with negligible speed; (b) Jumps into the disc, at its edge, with 9 m/s, directed towards the center; (c) Jumps into the disc s edge with 9 m/s, directed tangentially up, as shown in red; (d) Jumps into the disc s edge with 9 m/s, directed tangentially down (green); Page 13
14 PRACTICE: JUMPING OUT OF A MOVING DISC PRACTICE: A 200 kg disc 2 m in radius spins around a perpendicular axis through its center, with a person on it, at 3 rad/s counterclockwise. The person has mass 70 kg, is at rest (relative to the disc, that is, spins with it) at the disc s edge, and can be treated as a point mass. If the person jumps tangentially out of the disc with 10 m/s (relative to the floor), as shown by the red arrow, what new angular speed will the disc have as a result? BONUS: If the person steps out into ice with negligible speed of his/her own, what speed would it have upon exiting? Page 14
15 EXAMPLE: SPINNING ON A STRING OF VARIABLE LENGTH EXAMPLE: A small object (red, m1 = 2 kg), is on a smooth table top and attached to a light string that runs through a hole in the table. The other end of the string attaches to a hanging weight (green, m2). When the small object is given some speed, it spins in a circular path around the hole, with the tension from the hanging weight providing the centripetal force that keeps it spinning. Suppose the small object spins at 120 RPM when it is a radial distance of 10 cm from the hole. (a) How fast, in RPM, would it spin if the radial distance was reduced (by pulling on the hanging weight) to 6 cm? (b) At this new RPM, what linear (tangential) speed would the small object have? (c) What mass m2 does the hanging weight need to have to maintain the small object spinning at the RPM found in (b)? Page 15
16 PRACTICE: SPINNING ON A STRING OF VARIABLE LENGTH PRACTICE: A small object (red, m) is on a smooth table top and attached to a light string that runs through a hole in the table. The other end of the spring attaches to a hanging weight (green, M). When the small object is given some speed, it spins in a circular path around the hole, with the tension from the hanging weight providing the centripetal force that keeps it spinning. If the object spins with angular speed ω when it is a distance R from the central role, what new angular speed (in terms of ω) does it have when this distance is halved? BONUS: What new mass does the hanging weight need, in terms of M, to support a circular path at the new speed? Page 16
17 PRACTICE: BIRD COLLIDES AGAINST ROTATING DOORS PRACTICE: Two rotating doors, each 4 kg in mass and 6 m long, are fixed to the same central axis of rotation, as shown above (top view). Suppose a 4 kg bird flying with 30 m/s horizontal collides against the door and stays stuck to it, at a point 50 cm from one end. Calculate the angular speed with which the system (doors + bird) spin together. 50 cm I = 1/3 m A 2 A = side that extends away from axis I = 1/12 m (A 2 + B 2 ) A and B = sides on face of rectangle Page 17
18 INTRO TO ANGULAR MOMENTUM Remember: If you have linear speed ( ), you have Linear Momentum p = m v [ kg * m / s ]  If you have rotational speed ( ), you have Angular Momentum L = [ kg * m 2 / s ]  Difference: Linear Momentum is absolute; Angular Momentum is relative to the Axis of Rotation like Torque!  Do NOT confuse Angular Momentum (L = Iw) with Moment of Inertia (I; angular equivalent of mass). EXAMPLE: A solid cylinder of mass M = 5 kg and radius R = 2 m rotates about a perpendicular axis through its center with 120 RPM. Calculate its angular momentum about its central axis. Page 18
19 PRACTICE: ANGULAR MOMENTUM / FIND MASS PRACTICE: When solid sphere 4 m in diameter spins around its central axis at 120 RPM, it has 1,000 kg m 2 / s in angular momentum. Calculate the sphere s mass. Page 19
20 PRACTICE: ANGULAR MOMENTUM / COMPOSITE DISC PRACTICE: A composite disc is built from a solid disc and a concentric, thickwalled hoop, as shown below. The inner disc has mass 4 kg and radius 2 m. The outer disc (thickwalled) has mass 5 kg, inner radius 2 m, and outer radius 3 m. The two discs spin together and complete one revolution every 3 s. Calculate the system s angular momentum about its central axis. Page 20
21 ANGULAR MOMENTUM OF A POINT MASS A Point Mass in a circular path has rotational speed (w) and a linear equivalent (v,tan), but only ONE type of motion.  Note that Linear Momentum and Rotational Momentum do NOT give you the same number!  This is because Linear Momentum is absolute, and Angular Momentum is relative it depends on the Axis!  So if you calculate Linear and Angular Momenta for a Point Mass, you get different equations and numbers: EXAMPLE: A small 2kg object spins horizontally around a vertical axis at a rate of 3 rad/s, maintaining a constant distance of 4 m to the axis. Calculate the object s (a) linear momentum, and (b) angular momentum about its central axis. L = is also used for Angular Momentum of an object in LINEAR motion, about an axis of rotation (more soon) Page 21
22 PRACTICE: ANGULAR MOMENTUM / EARTH PRACTICE: The Earth has mass 5.97 x kg, radius 6.37 x 10 6 m. The EarthSun distance is 1.5 x m. Calculate its angular momentum as it spins around itself. Treat the Earth as a solid sphere of uniform mass distribution. BONUS 1: Treating the Earth as a point mass, calculate its angular momentum as it spins around the Sun. BONUS 2: Does the Earth have linear momentum as it spins around (i) itself; (ii) the Sun? Page 22
23 PRACTICE: ANGULAR MOMENTUM / ROD WITH MASSES PRACTICE: A system is made of two small, 3 kg masses attached to the ends of a 5 kg, 4 m long, thin rod. The system rotates with 180 RPM about an axis perpendicular to the rod and through one of its ends, as shown. Calculate the system s angular momentum about its axis. Page 23
24 ANGULAR MOMENTUM OF OBJECTS IN LINEAR MOTION In some problems, an object moving in a straight line collides against an object fixed in a rotating axis:  Remember we used Linear Momentum to solve Collision problems!  BUT in this case, we need the first object s Momentum, not its Momentum.  But how do you get the Momentum of an object that is moving in a straight line?!?  An object in a straight line has Angular Momentum relative to unrelated axis of rotation L =  Notice that this is the SAME equation as the Angular Momentum of a. EXAMPLE: Two rotating doors, each 6.0 m long, are fixed to the same central axis of rotation, as shown above (top view). Suppose a 4 kg bird flying with 30 m/s horizontal is about to collide against the door, at a point 50 cm from one end. Calculate the bird s angular momentum about the axis through the center of the door, just before hitting the door. Later we will see how to FULLY solve these types of Rotational Collisions! (For now, contain your excitement, please!) Page 24
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