Physics I 95.141 LECTURE 18 11/15/10

Department of Physics and Applied Physics95.141, F2010, Lecture 18

Physics I95.141

LECTURE 1811/15/10


Outline/Notes

• Outline– Center of Mass– Angular quantities– Vector nature of

angular quantities– Constant angular

acceleration

• Administrative Notes– HW review session

moved to Thursday, 11/18, 6:30pm in OH218.


Review• In the previous lecture we discussed collisions in 2D and 3D.

– Momentum always conserved! Can write a conservation of momentum expression for each dimension/component.

– If the collision is elastic, then we can also say that Kinetic Energy is conserved, and include this in our equations:

• We also discussed the Center of Mass (CM)– Calculation of CM for 1D point masses

– Calculation of CM for 3D point masses

– Calculation of CM for symmetric solid objects

systemsystem pp

22

2

1

2

1iiii vmvm

xx pp yy pp zz pp


Solid Objects• We can easily find the CM for a collection of point

masses, but most everyday items aren’t made up of 2 or 3 point masses. What about solid objects?

• Imagine a solid object made out of an infinite number of point masses. The easiest trick we can use is that of symmetry!


Solid Objects (General)• If symmetry doesn’t work, we can solve for CM

mathematically. – Divide mass into smaller sections dm.

rdm


Solid Objects (General)

• If symmetry doesn’t work, we can solve for CM mathematically. – Divide mass into smaller sections dm.

xdmM

xCM1

ydmM

yCM1

zdmM

zCM1

i

iiCM dmxM

x1


Example: Rod of varying density

• Imagine we have a circular rod (r=0.1m) with a mass density given by ρ=2x kg/m3.

L=2mx


Example: Rod of varying density

• Imagine we have a circular rod (r=0.1m) with a mass density given by ρ=2x kg/m3.

L=2mx


CM and Translational Motion

• The translational motion of the CM of an object is directly related to the net Force acting on the object.

• The sum of all the Forces acting on the system is equal to the total mass of the system times the acceleration of its center of mass.

• The center of mass of a system of particles (or objects) with total mass M moves like a single particle of mass M acted upon by the same net external force.

extCM FaM


Example• A 60kg person stands on the right most edge of a uniform board of

mass 30kg and length 6m, lying on a frictionless surface. She then walks to the other end of the board. How far does the board move?


CM Review• What is the center of mass of the shape below, if we assume a constant surface density (σ

[kg/m2])?

1m

1m

6m

4m(0,0)

4m

1m


CM Review

• Calculate motion of the letter K (total mass MK=2kg) if a Force is applied to the letter.

iF ˆ4


Motion of an object/system under a Force

• We know that for a system of masses, or for a solid object, if a Force is applied to the system/object, the center of mass of the moves as if all of the mass was at the CM and the Force is applied to the CM.• But does this entirely determine the motion of the object?


Rotation

• Objects don’t only move translationally, but can also vibrate or rotate.

• In this chapter (10) we are going to look at rotational motion.

• First, we need to go back and review the nomenclature we use to describe rotational motion.

• Motion of an object can be described by translational motion of the CM + rotation of the object around its CM!


Circular Motion Nomenclature: Angular Position

• It is easiest to describe circular motion in polar coordinates.

y

x

R

Rarclength

,R

R

Axis of rotation

For θ in radians!!!


Circular Motion Nomenclature: Angular Displacement

• Angular displacement

12

R

Axis of rotation Axis of rotation


Circular Motion Nomenclature: Angular Velocity and Acceleration

• Average Angular Velocity

• Instantaneous Angular Velocity

• Average Angular acceleration

• Instantaneous Angular acceleration

12

12

ttt

dt

d

tt

0

lim

dt

d

tt

0

lim

12

12

ttt


Usefulness of Angular Quantities

• Each point on a rotating rigid body has the same angular displacement, velocity, and acceleration!

• What about translational quantities?

dt

dv

Rdd


Tangential Acceleration

• If we can calculate tangential velocity from angular velocity and radius:

• We can also calculate tangential acceleration:

• So, total acceleration is:

Rv tan

Rdt

dva tan

tan

Rtotal aaa

tan


Example• A record (r=15cm), starting from rest, accelerates with a constant

angular acceleration α=0.2 rad/s for 5 seconds. What is (a) the angular velocity of the record at t=5s? (b) the linear velocity of a point on the edge of the record (t=5s)? (c) and the linear and centripetal acceleration of a point on the edge of the record (t=5s)?


Frequency and Period

• We can relate the angular velocity of rotation to the frequency of rotation:

• Can also write the period in terms of angular velocity, but Period (T) only makes sense for uniform circular motion.


Vector Nature of Angular Quantities

• We can treat both ω and α as vectors• If we look at points on the wheel, they all have different velocities in the xy plane

– Choosing a vector in the xy plane doesn’t make sense– Choose vector in direction of axis of rotation– But which direction?

z

•Right Hand Rule•Use fingers on right hand to trace rotation of object•Direction thumb points is vector direction for angular velocity, acceleration


Constant Angular Acceleration

• In Chapter 2, we discussed the kinematic equations for motion with constant acceleration.

2

)(2

2

1

22

2

o

oo

oo

o

vvv

xxavv

attvxx

atvv


Rotational vs. Translational Equations of Motion

• The equations of motion for translational motion and rotational motion are parallel!– Makes it very easy to remember!

2

2

1ttoo

2o

2

)(2

2

1

22

2

o

oo

oo

o

vvv

xxavv

attvxx

atvv

oo 222

to


Constant Angular Acceleration

• If you can remember your kinematic equations for translational motion, you can solve problems with constant angular acceleration!

va

x

tt


Example• A top is brought up to speed with α=7rad/s2 in

1.5s. After that it slows down slowly with α=-0.1rad/s2 until it stops spinning. – A) What is the fastest angular velocity of the top?– B) How long does it take the top to stop spinning once

it reaches its top angular velocity?– C) How many rotations does the top make in this time?


Example• A top is brought up to speed with α=7rad/s2 in

1.5s. After that it slows down slowly with α=-0.1rad/s2 until it stops spinning. – C) How many rotations does the top make in this time?


What Did We Learn Today?

• Center of Mass– Symmetry– Integration– Translational Motion of…

• Angular Motion– Nomenclature for angular motion

• Angular displacement• Angular velocity• Angular acceleration

– Constant angular acceleration• Symmetry with equations of translational motion

Physics I 95.141 LECTURE 18 11/15/10

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