physics 218 lecture 23

Physics 218, Lecture XXIII

1

Physics 218Lecture 23

Dr. David Toback

Physics 218, Lecture XXIII 2

Checklist for Today•Things due Monday

–Chapter 14 in WebCT•Things that were due yesterday

–Chapter 15 problems as Recitation Prep

•Things due next Monday–Chapter 15 & 16 in WebCT

•Next week–Chapter 18 reading


The ScheduleThis Week (4/14)• Monday: Chapter 14 due in WebCT• Tues: Exam 3 (Chaps 10-13)• Wed: Recitation on Chap 15, Lab• Thurs: Last lecture on Chaps 12-16Next Week (4/21)• Monday: Chapter 15 & 16 due in WebCT• Tues:

– Reading for Chapter 18– Lecture on Chapter 18

• Wed: Recitation on Chapter 18• Thurs: Last lecture, Chapter 18Week after that (4/28)• No lectures or recitationsWeek after that (5/5) • Final: Monday May 5th, 1PM-3PM in this room


Overview

•Chapters 12-16 are about Rotational Motion

•Concentrate on the relationship between linear and angular variables

•Do them in six combined lectures – Last lecture today– Angular Momentum and Energy

•The book does the math, I’ll focus on the understanding and making the issues more intuitive


Angular Quantities

• Position Angle • Velocity Angular Velocity • Acceleration Angular

Acceleration • Force Torque • Mass Moment of Inertia Today we’ll finish:

–Momentum–Energy


Momentum

Momentum vs. Angular Momentum:

Newton’s Laws:

ILvmp

dtLd

dtpd

F


Angular Momentum Definition

Another definition:

p r L


Angular Motion of a Particle

Determine the angular momentum, L, of a particle, with mass m and speed v, moving in circular motion with radius r


Conservation of Angular Momentum

Const L 0 ifdtLd

By Newton’s laws, the angular momentum of a body can change, but the angular momentum for a system

cannot change


Same as for linear momentum


Ice Skater

• This one you’ve seen on TV

• Try this at home in a chair that rotates

• Get yourself spinning with your arms and legs stretched out, then pull them in

IL


Problem Solving

For Conservation of Angular Momentum problems:

BEFORE and AFTER



Before



After


Clutch DesignAs a car engineer, you model a car clutch as two plates, each with radius R, and masses MA and MB (IPlate = ½MR2). Plate A spins with speed 1 and plate B is at rest. You close them so they spin together

Find the final angular velocity of the system


Angular Quantities

• Position Angle • Velocity Angular Velocity • Acceleration Angular Acceleration • Force Torque • Mass Moment of Inertia Today we’ll finish:

– Momentum Angular Momentum L

– Energy


Rotational Kinetic Energy

KEtrans = ½mv2

KErotate = ½I2

Conservation of Energy must take rotational kinetic energy

into account


Rotation and Translation• Objects can both Rotate

and Translate

• Need to add the two

KEtotal = ½ mv2 + ½I2

• Rolling without slipping is a special case where you can relate the two

V = r


Rolling Down an InclineYou take a solid ball of mass m and radius R

and hold it at rest on a plane with height Z. You then let go and the ball rolls without slipping.

What will be the speed of the ball at the bottom?

What would be the speed if the ball didn’t roll and there were no friction?Note:

Isphere = 2/5MR2

Z


A bullet strikes a cylinder

A bullet of speed V and mass m strikes a solid cylinder of mass M and inertia I=½MR2, at radius R and sticks. The cylinder is anchored at point 0 and is initially at rest.

What is of the system after the collision?

Is energy Conserved?


Rotating RodA rod of mass

uniform density, mass m and length l pivots at a hinge. It has moment of inertia I=ml/3 and starts at rest at a right angle. You let it go:

What is when it reaches the bottom?

What is the velocity of the tip at the bottom?


Less Spherical Heavy PulleyA heavy pulley, with radius

R, starts at rest. We pull on an attached rope with constant force FT. It accelerates to final angular speed in time t.

A better estimate takes into account that there is friction in the system. This gives a torque (due to the axel) we’ll call this fric.

What is this better estimate of the moment of Inertia?

R


Person on a DiskA person with mass

m stands on the edge of a disk with radius R and moment ½MR2. Neither is moving.

The person then starts moving on the disk with speed V.

Find the angular velocity of the disk


Same Problem: Forces

Same problem but with Forces


Challenge Exam• Announcement of this semester’s 218

Challenge Exam: Monday April 28th at 6:00PM– Not required (just for fun)– Does not (will not!!!) count as part of

your final grade• Test your skills against the best Aggies from

all the Physics 218 sections (not just this lecture) on Physics 218 material

• Students who perform well will be recognized as Mechanics Scholars, and honored at a banquet in their honor. Other prizes, including cash.

• Handout information on my WebSite


Next Week• Reading: Chapter 18 “Harmonic

Motion”– Last topic for the course

Notes for the final:• No more “bonus” points• Exam is worth 200 points• Problem 8 will consist of two items:

– 5 points for getting 100% on all your homework & WebCT quizzes

– 5 more points for getting 100 on the Mini-practice Exam final


Spherical Heavy Pulley

A heavy pulley, with radius R, starts at rest. We pull on an attached rope with a constant force FT. It accelerates to an angular speed of in time t.

What is the moment of inertia of the pulley?

R


Less Spherical Heavy Pulley

A heavy pulley, with radius R, starts at rest. We pull on an attached rope with constant force FT. It accelerates to final angular speed in time t.

A better estimate takes into account that there is friction in the system. This gives a torque (due to the axel) we’ll call this fric.

What is this better estimate of the moment of Inertia?

R


Exam II

• Mean = 75– Please check to make sure they

added your points correctly AND entered them into WebCT correctly!!!

• Average on first two exam = 76%– Straight scale so far…

• Reading quizzes should be passed back in recitation


Next Time• Chapter 11

– Reading Questions: Q11.X & Q11.X– XXX FIXME!!!– Math, Torque, Angular Momentum,

Energy again, but more sophisticated– The material will not be on the 3rd

exam, but will help with the exam. It will all be on the final

• HW 10 Due Monday• Exam 3 is next Thursday, April 22nd


Angular Quantities

• Position Angle • Velocity Angular Velocity • Acceleration Angular Acceleration • Force Torque Today we’ll finish:

– Mass– Momentum– Energy


Calculating Moments of Inertia

dmr I

mr I

2

2

Here r is the distance from the axis of each little piece of mass


Calculate the Moment of Inertia

A pulley has mass M, uniform density, radius R, and rotates around its fixed axis

Calculate its moment of inertia

R


Calculate the Moment of Inertia

Better example here…

Calculate its moment of inertia

R


Hollow Cylinder

Consider a hollow cylinder with uniform density, inner radius R1, outer radius R2 and total Mass M.Find the moment of Inertia


Parallel-Axis Theorem

• Quick Trick for calculating Moments

• I = Icm + Mh2

• Example


• Old stuff


Kepler’s 2nd Law

2nd Law: Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out area in equal periods of time


Atwood’s MachineA pulley with a fixed center (at point O), radius R and moment of inertia I, has a massless rope wrapped around it (no slipping). The rope has two masses, m1 and m2 attached to its ends. Assume m2>m1

•What is the acceleration of the system?

•Do some checks.


Why does the Bicycle Wheel Turn to the Right?


Angular Momentum

Again we use the Cross Product:

Derivation of = dL/dt

p r

L


L for a system of many bodies

• Have to be careful with Angular Momentum– = dl/dt for a single particle– = dl/dt) for a system of many

particles– All internal torques cancel because of

Newton’s law (all internal forces are equal and opposite)

• Reference Frame matters. Only true for:– The origin is an inertial Reference Frame– The center of mass


L for a Rigid Body

Find the angular momentum, L, for this body given that it is rotating around the Z axis with angular velocity

physics 218 lecture 23

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