Goals:

1
Lecture 17

• Goals

• Chapter 12
• Define center of mass
• Analyze rolling motion
• Use Work Energy relationships
• Introduce torque
• Equilibrium of objects in response to forces
  torques

• Assignment
• HW7 due tomorrow
• Wednesday, Exam Review

2
Combining translation and rotation

• Objects can have translational energy
• Objects can have rotational energy
• Objects can have both K ½ m v2 ½ I w2

3
1st A special point for rotationCenter of Mass
(CM)

• A free object will rotate about its center of
  mass.
• Center of mass Where the system is balanced !
• A mobile exploits this centers of mass.

mobile
4
System of Particles Center of Mass

• How do we describe the position of a system
  made up of many parts ?
• Define the Center of Mass (average position)
• For a collection of N individual point like
  particles whose masses and positions we know

RCM
m2
m1
r2
r1
y
x
(In this case, N 2)
5
Sample calculation
m at ( 0, 0) 2m at (12,12) m at (24, 0)

• Consider the following mass distribution

XCM (m x 0 2m x 12 m x 24 )/4m meters YCM
(m x 0 2m x 12 m x 0 )/4m meters XCM 12
meters YCM 6 meters
6
System of Particles Center of Mass

• For a continuous solid, one can convert sums to
  an integral.

dm
r
y
x
where dm is an infinitesimal mass element but
there is no new physics.
7
Connection with motion...

• An unconstrained rigid object with rotation and
  translation rotates about its center of mass!
• Any point p rotating

8
Work Kinetic Energy

• Work Kinetic-Energy Theorem ?K WNET
• Applies to both rotational as well as linear
  motion.
• What if there is rolling without slipping ?

9
Same Example Rolling, without slipping, Motion

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

10
Rolling without slipping motion

• Again consider a cylinder rolling at a constant
  speed.

2VCM
CM
VCM
11
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
If acceleration acenter of mass - aR
12
Example Rolling Motion

• A solid 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 ICM w2 and v wR Mgh ½
Mv2 ½ (½ M R2 )(v/R)2 ¾ Mv2 v 2(gh/3)½
13
How do we reconcile force, angular velocity and
angular acceleration?
14
From force to spin (i.e., w) ?
A force applied at a distance from the rotation
axis gives a torque
a
FTangential
F
q
Fradial
r
r
FTangential
Fradial
FTang sin q

• If a force points at the axis of rotation the
  wheel wont turn
• Thus, only the tangential component of the force
  matters
• With torque the position angle of the force
  matters

• ?NET r FTang r F sin q

15
Rotational Dynamics What makes it spin?

• ?NET r FTang r F sin q

• 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
16
Torque

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

?NET I a

• This is the rotational version of FNET ma
  
• Moment of inertia, I Si mi ri2 , is the
  rotational equivalent of mass.
• If I is big, more torque is required to achieve
  a given angular acceleration.

17
Rotational Dynamics
a
FTangential
F
Fradial

• ?NET I a r F sin q

r

• A constant torque gives constant angular
  acceleration if and only if the mass distribution
  and the axis of rotation remain constant.

18
Torque, like w, has pos./neg. values

• Magnitude is given by (1) r F sin q
• (2) Ftangential r
• (3) 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

r sin q
line of action
F cos(90-q) FTang.
r
a
90-q
q
F
F
F
Fradial
r
r
r
19
Statics
Equilibrium is established when
In 3D this implies SIX expressions (x, y z)
20
Example

• Two children (30 kg 60 kg) sit on a horizontal
  teeter-totter. The larger child is at the end of
  the bar and 1.0 m from the pivot point. The
  smaller child is trying to figure out where to
  sit so that the teeter-totter remains horizontal
  and motionless. The teeter-totter is a uniform
  bar of length 3.0 m and mass 30 kg.
• Assuming you can treat both children as point
  like particles, what is the initial angular
  acceleration of the teeter-totter when the large
  child lifts up their legs off the ground (the
  smaller child cant reach)?
• The moment of inertia of the bar about the pivot
  is 30 kg m2.
• For the static case

21
Example Soln.
30 kg

• Draw a Free Body diagram (assume g 10 m/s2)
• 0 300 d 300 x 0.5 N x 0 600 x 1.0
• 0 2d 1 4
• d 1.5 m from pivot point

22
Recap

• Assignment
• HW7 due tomorrow
• Wednesday review session