Lecture 17
This presentation is the property of its rightful owner.
Sponsored Links
1 / 22

Lecture 17 PowerPoint PPT Presentation


  • 45 Views
  • Uploaded on
  • Presentation posted in: General

Lecture 17. Chapter 12 U nderstand the equilibrium dynamics of an extended object in response to forces Analyze rolling motion Employ “conservation of angular momentum” concept. Goals:. Assignment: HW7 due March 25 th After Spring Break Tuesday: Catch up.

Download Presentation

Lecture 17

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Lecture 17

Lecture 17

  • Chapter 12

    • Understand the equilibrium dynamics of an extended object in response to forces

    • Analyze rolling motion

    • Employ “conservation of angular momentum” concept

Goals:

Assignment:

  • HW7 due March 25th

  • After Spring Break Tuesday:

    Catch up


Rotational dynamics what makes it spin

Rotational Dynamics: What makes it spin?

  • A “special” net force is necessary and that action depends on the location from the axis of rotation

  • For translational motion and acceleration the position of the force doesn’t matter (doesn’t change the physics we see)

  • IMPORTANT: For rotational motion and angular acceleration: we always reference the specific position of the force relative to the axis of rotation.

  • Vectors are simply a tool for visualizing Newton’s Laws


Rotational dynamics what makes it spin1

Rotational Dynamics: What makes it spin?

A force applied at a distance from the rotation axis gives a torque

a

FTangential

NET = |r| |FTang| ≡|r||F| sin f

F

Fradial

r

  • Only the tangential component of the force matters

  • With torque the position of the force matters


Rotational dynamics what makes it spin2

a

FTangential

F

Fradial

r

Rotational Dynamics: What makes it spin?

A force applied at a distance from the rotation axis

NET = |r| |FTang| ≡|r||F| sin f

  • Torque is the rotational equivalent of force

    Torque has units of kg m2/s2 = (kg m/s2) m = N m

NET = r FTang = r m aTang

= r m r a

= (m r2) a

For every little part of the wheel


For a point mass net m r 2 a and inertia

a

FTangential

F

Frandial

r

For a point massNET = m r2a and inertia

The further a mass is away from this axis the greater the inertia (resistance) to rotation

  • This is the rotational version of FNET = ma

  • Moment of inertia, I≡ m r2 , (here I is just a point on the wheel) is the rotational equivalent of mass.

  • If I is big, more torque is required to achieve a given angular acceleration.


Rotational dynamics what makes it spin3

Rotational Dynamics: What makes it spin?

A force applied at a distance from the rotation axis gives a torque

a

FTangential

NET = |r| |FTang| ≡|r||F| sin f

F

Fradial

r

  • A constant torque gives constant angularacceleration iff the mass distribution and the axis of rotation remain constant.


Angular motion can be described by vectors

Angular motion can be described by vectors

  • Recall that linear motion involves x, y and/or z vectors

  • Angular motion can be quantified by defining a vector along

    the axis of rotation.

  • The axis of rotation is the one set of points that is fixed with respect to a rotation.

  • Use the right hand rule


Torque is a vector quantity

F cos(90°-q) = FTang.

line of action

r

a

90°-q

q

F

F

r sin q

F

Fradial

r

r

r

Torque is a vector quantity

  • Magnitude is given by |r| |F| sin q

    or, equivalently, by the |Ftangential | |r|

    or by |F| |rperpendicular to line of action |

  • Direction is parallel to the axis of rotation with respect to the “right hand rule”

  • And for a rigid object= I a


Work kinetic energy

Work & Kinetic Energy:

  • Recall the Work Kinetic-Energy Theorem: K = WNET

  • This applies to both rotational as well as linear motion.

  • So for an object that rotates about a fixed axis

  • For an object which is rotating and translating


Exercise torque magnitude

Exercise Torque Magnitude

  • Case 1

  • Case 2

  • Same

  • In which of the cases shown below is the torque provided by the applied force about the rotation axis biggest? In both cases the magnitude and direction of the applied force is the same.

  • Remember torque requires F, rand sin q

    or the tangential force component times perpendicular distance

L

F

F

L

axis

case 1

case 2


Exercise torque magnitude1

L

F

F

L

axis

case 1

case 2

Exercise Torque Magnitude

  • In which of the cases shown below is the torque provided by the applied force about the rotation axis biggest? In both cases the magnitude and direction of the applied force is the same.

  • Remember torque requires F,rand sin f

    or the tangential force component times perpendicular distance

(A)case 1

(B)case 2

(C) same


Example rotating rod

L

m

Example: Rotating Rod

  • A uniform rod of length L=0.5 m and mass m=1 kg is free to rotate on a frictionless pin passing through one end as in the Figure. The rod is released from rest in the horizontal position. What is

    (A) its angular speed when it reaches the lowest point ?

    (B) its initial angular acceleration ?

    (C) initial linear acceleration of its free end ?


Example rolling motion

Ball has radius R

M

M

M

M

M

h

M

v ?

q

M

Example :Rolling Motion

  • A cylinder is about to roll down an inclined plane. What is its speed at the bottom of the plane ?


Example rolling motion1

Ball has radius R

M

M

M

M

M

h

M

v ?

q

M

Example :Rolling Motion

  • A cylinder is about to roll down an inclined plane. What is its speed at the bottom of the plane ?

  • Use Work-Energy theorem

Mgh = ½ Mv2 + ½ ICMw2

Mgh = ½ Mv2 + ½ (½ M R2)(v/R)2 = ¾ Mv2

v = 2(gh/3)½


Motion

Motion

  • Again consider a cylinder rolling at a constant speed.

Both with

|VTang| = |VCM |

Rotation only

VTang = wR

Sliding only

2VCM

VCM

CM

CM

CM

VCM


Angular momentum

Angular Momentum:

  • We have shown that for a system of particles,

    momentum

    is conserved if

  • What is the rotational equivalent of this (rotational “mass” times rotational velocity)?

    angular momentum

    is conserved if


Angular momentum of a rigid body about a fixed axis

v1

m2

j

m1

r2

r1

i

v2

r3

v3

m3

Angular momentum of a rigid body about a fixed axis:

  • Consider a rigid distribution of point particles rotating in the x-y plane around the z axis, as shown below. The total angular momentum around the origin Is the sum of the angular momentum of each particle:

  • Even if no connecting rod we can deduce an Lz

( ri and vi, are perpendicular)

Using vi =  ri, we get


Example two disks

z

z

F

Example: Two Disks

  • A disk of mass M and radius R rotates around the z axis with angular velocity 0. 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 F.

0


Example two disks1

z

z

0

F

Example: Two Disks

  • A disk of mass M and radius R rotates around the z axis with initial angular velocity 0. A second identical disk, at rest, is dropped on top of the first. There is friction between the disks, and eventually they rotate together with angular velocity F.

No External Torque so Lz is constant

Li = Lf I wii = I wf½ mR2w0 = ½ 2mR2wf


Example throwing ball from stool

Example: Throwing ball from stool

  • A student sits on a stool, initially at rest, but which is free to rotate. The moment of inertia of the student plus the stool is I. They throw a heavy ball of mass M with speed v such that its velocity vector moves a distance d from the axis of rotation.

    • What is the angular speed F of the student-stool system after they throw the ball ?

M

v

F

d

I

I

Top view: before after


Angular momentum as a fundamental quantity

Angular Momentum as a Fundamental Quantity

  • The concept of angular momentum is also valid on a submicroscopic scale

  • Angular momentum has been used in the development of modern theories of atomic, molecular and nuclear physics

  • In these systems, the angular momentum has been found to be a fundamental quantity

    • Fundamental here means that it is an intrinsic property of these objects


Lecture 171

Lecture 17

Assignment:

  • HW7 due March 25th

  • Thursday: Review session


  • Login