Download Lecture 18

Physics I
95.141
LECTURE 18
11/15/10
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
Outline/Notes
• Administrative Notes
– HW review session
moved to Thursday,
11/18, 6:30pm in
OH218.
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
• Outline
– Center of Mass
– Angular quantities
– Vector nature of
angular quantities
– Constant angular
acceleration
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.


psystem  psystem
 p   p  p   p  p   p
x
x
y
y
z
z
– If the collision is elastic, then we can also say that Kinetic Energy is
conserved, and include this in our equations:
1
1
2
2

m
v

m
v
2 i i 2 i i
• 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
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
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!
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
Solid Objects (General)
• If symmetry doesn’t work, we can solve for CM
mathematically.
– Divide mass into smaller sections dm.
dm

r
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
Solid Objects (General)
• If symmetry doesn’t work, we can solve for CM
mathematically.
– Divide mass into smaller sections dm.
xCM
xCM 
1
xdm

M
1

M
 x dm
i
i
i
yCM 
1
M
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
 ydm
zCM 
1
zdm

M
Example: Rod of varying density
• Imagine we have a circular rod (r=0.1m) with a mass
density given by ρ=2x kg/m3.
x
L=2m
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
Example: Rod of varying density
• Imagine we have a circular rod (r=0.1m) with a mass
density given by ρ=2x kg/m3.
x
L=2m
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
CM and Translational Motion
• The translational motion of the CM of an object is directly
related to the net Force acting on the object.


MaCM   Fext
• 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.
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
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?
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
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
1m
(0,0)
4m
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
CM Review
• Calculate motion of the letter K (total mass MK=2kg) if a
Force is applied to the letter.

F  4iˆ
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
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?
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
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!
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
Circular Motion Nomenclature:
Angular Position
• It is easiest to describe circular motion in polar
coordinates.
R, 
y
R

  arclength
  R For θ in radians!!!

R
x
Axis of rotation
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
Circular Motion Nomenclature:
Angular Displacement
• Angular displacement
  2  1
R
Axis of rotation
Axis of rotation

95.141, F2010, Lecture 18
Department of Physics and Applied Physics
Circular Motion Nomenclature:
Angular Velocity and Acceleration
• Average Angular Velocity
  2  1


t
t 2  t1
• Instantaneous Angular Velocity
lim  d


t  0 t
dt
• Average Angular acceleration
 2  1
 

t
t 2  t1
• Instantaneous Angular acceleration
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
lim  d


t  0 t
dt
Usefulness of Angular Quantities
• Each point on a rotating rigid body has the same
angular displacement, velocity, and acceleration!
• What about translational quantities?
d
v
d  Rd
dt
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
Tangential Acceleration
• If we can calculate tangential velocity from
angular velocity and radius:
vtan  R
• We can also calculate tangential acceleration:
dv tan
a tan 
 R
dt
• So, total acceleration is:



atotal  atan  a R
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
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)?
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
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.
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
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
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
Constant Angular Acceleration
• In Chapter 2, we discussed the kinematic
equations for motion with constant acceleration.
v  vo  at
1 2
x  xo  vot  at
2
2
2
v  vo  2a ( x  xo )
v  vo
v
2
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
Rotational vs. Translational Equations of Motion
• The equations of motion for translational motion
and rotational motion are parallel!
– Makes it very easy to remember!
  o  t
1 2
   o   o t  t
2
 2  o2  2    o 

  o
2
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
v  vo  at
1 2
x  xo  vot  at
2
2
2
v  vo  2a ( x  xo )
v  vo
v
2
Constant Angular Acceleration
• If you can remember your kinematic equations
for translational motion, you can solve problems
with constant angular acceleration!
x
 v
 a
tt
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
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?
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
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?
95.141, F2010, Lecture 18
Department of Physics and Applied Physics
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
95.141, F2010, Lecture 18
Department of Physics and Applied Physics