Unformatted text preview:

Angular Momentum; General RotationUnits of Chapter 11Slide 311-1 Angular Momentum—Objects Rotating About a Fixed AxisSlide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 1211-2 Vector Cross Product; Torque as a VectorSlide 14Slide 15Slide 16Slide 17Slide 1811-3 Angular Momentum of a ParticleSlide 20Slide 2111-4 Angular Momentum and Torque for a System of Particles; General Motion11-5 Angular Momentum and Torque for a Rigid ObjectSlide 24Slide 25Slide 26Slide 2711-6 Conservation of Angular MomentumSlide 29Slide 30Chapter 11Angular Momentum; General Rotation•Angular Momentum—Objects Rotating About a Fixed Axis•Vector Cross Product; Torque as a Vector•Angular Momentum of a Particle•Angular Momentum and Torque for a System of Particles; General Motion•Angular Momentum and Torque for a Rigid ObjectUnits of Chapter 11•Conservation of Angular Momentum•The Spinning Top and Gyroscope•Rotating Frames of Reference; Inertial Forces•The Coriolis EffectUnits of Chapter 1111-1 Angular Momentum—Objects Rotating About a Fixed AxisThe rotational analog of linear momentum is angular momentum, L:Then the rotational analog of Newton’s second law is:This form of Newton’s second law is valid even if I is not constant.11-1 Angular Momentum—Objects Rotating About a Fixed AxisIn the absence of an external torque, angular momentum is conserved:More formally,the total angular momentum of a rotating object remains constant if the net external torque acting on it is zero.0 and constant.dLL Idtw= = =11-1 Angular Momentum—Objects Rotating About a Fixed AxisThis means:Therefore, if an object’s moment of inertia changes, its angular speed changes as well.11-1 Angular Momentum—Objects Rotating About a Fixed AxisExample 11-1: Object rotating on a string of changing length. A small mass m attached to the end of a string revolves in a circle on a frictionless tabletop. The other end of the string passes through a hole in the table. Initially, the mass revolves with a speed v1 = 2.4 m/s in a circle of radius R1 = 0.80 m. The string is then pulled slowly through the hole so that the radius is reduced to R2 = 0.48 m. What is the speed, v2, of the mass now?11-1 Angular Momentum—Objects Rotating About a Fixed AxisExample 11-2: Clutch.A simple clutch consists of two cylindrical plates that can be pressed together to connect two sections of an axle, as needed, in a piece of machinery. The two plates have masses MA = 6.0 kg and MB = 9.0 kg, with equal radii R0 = 0.60 m. They are initially separated. Plate MA is accelerated from rest to an angular velocity ω1 = 7.2 rad/s in time Δt = 2.0 s. Calculate (a) the angular momentum of MA, and (b) the torque required to have accelerated MA from rest to ω1. (c) Next, plate MB, initially at rest but free to rotate without friction, is placed in firm contact with freely rotating plate MA, and the two plates both rotate at a constant angular velocity ω2, which is considerably less than ω1. Why does this happen, and what is ω2?11-1 Angular Momentum—Objects Rotating About a Fixed AxisExample 11-3: Neutron star.Astronomers detect stars that are rotating extremely rapidly, known as neutron stars. A neutron star is believed to form from the inner core of a larger star that collapsed, under its own gravitation, to a star of very small radius and very high density. Before collapse, suppose the core of such a star is the size of our Sun (r ≈ 7 x 105 km) with mass 2.0 times as great as the Sun, and is rotating at a frequency of 1.0 revolution every 100 days. If it were to undergo gravitational collapse to a neutron star of radius 10 km, what would its rotation frequency be? Assume the star is a uniform sphere at all times, and loses no mass.11-1 Angular Momentum—Objects Rotating About a Fixed AxisAngular momentum is a vector; for a symmetrical object rotating about a symmetry axis it is in the same direction as the angular velocity vector.11-1 Angular Momentum—Objects Rotating About a Fixed AxisExample 11-4: Running on a circular platform.Suppose a 60-kg person stands at the edge of a 6.0-m-diameter circular platform, which is mounted on frictionless bearings and has a moment of inertia of 1800 kg·m2. The platform is at rest initially, but when the person begins running at a speed of 4.2 m/s (with respect to the Earth) around its edge, the platform begins to rotate in the opposite direction. Calculate the angular velocity of the platform.11-1 Angular Momentum—Objects Rotating About a Fixed AxisConceptual Example 11-5: Spinning bicycle wheel.Your physics teacher is holding a spinning bicycle wheel while he stands on a stationary frictionless turntable. What will happen if the teacher suddenly flips the bicycle wheel over so that it is spinning in the opposite direction?11-2 Vector Cross Product; Torque as a VectorThe vector cross product is defined as:The direction of the cross product is defined by a right-hand rule:11-2 Vector Cross Product; Torque as a VectorThe cross product can also be written in determinant form:11-2 Vector Cross Product; Torque as a VectorSome properties of the cross product:11-2 Vector Cross Product; Torque as a VectorTorque can be defined as the vector product of the force and the vector from the point of action of the force to the axis of rotation:11-2 Vector Cross Product; Torque as a VectorFor a particle, the torque can be defined around a point O:Here, is the position vector from the particle relative to O.rrExample 11-6: Torque vector.Suppose the vector is in the xz plane, and is given by = (1.2 m) + 1.2 m) Calculate the torque vector if = (150 N) .11-2 Vector Cross Product; Torque as a VectorrtrrFrrr11-3 Angular Momentum of a ParticleThe angular momentum of a particle about a specified axis is given by:11-3 Angular Momentum of a ParticleIf we take the derivative of , we find:Sincewe have:rL11-3 Angular Momentum of a ParticleConceptual Example 11-7: A particle’s angular momentum.What is the angular momentum of a particle of mass m moving with speed v in a circle of radius r in a counterclockwise direction?11-4 Angular Momentum and Torque for a System of Particles; General MotionThe angular momentum of a system of particles can change only if there is an external torque—torques due to internal forces cancel.This equation is valid in any inertial reference frame. It is also valid for the center of mass, even if it is accelerating:11-5 Angular Momentum and Torque for a Rigid ObjectFor a


View Full Document

TTU PHYS 1408 - Angular Momentum;

Download Angular Momentum;
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Angular Momentum; and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Angular Momentum; 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?