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CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 17
Course website: faculty.uml.edu/pchowdhury/95.141/
www.masteringphysics.com
Course: UML95141SPRING2013
Lecture Captureh"p://echo360.uml.edu/chowdhury2013/physics1Spring.html
95.141 Apr 3 , 2013 PHYSICS I Lecture 17
Last Lecture Today
Chapter 9 2-D collisions Systems of particles (extended objects) Center of mass
Chapter 10 Rotational Motion Rotational kinematics Rotational dynamics Torque
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 17
Center of Mass: Wrap-up
A 50 kg person stands on the right most edge of a uniform board of mass 25 kg and length 6 m, lying on a frictionless surface. She then walks to the other end of the board. How far does the board move?
50 kg 25 kg
25 kg 50 kg
c.m.
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 17
(m1 +m2 )xCM =m1x1 +m2x2
6m
3m 2m
1m
?
Rotational Motion In addition to translation, extended objects can rotate
Need to develop a vocabulary for
describing rotational motion
There is rotation everywhere you look in the universe, from the nuclei of atoms to spiral galaxies
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 17
Angular Position Easiest to describe rotation in polar coordinates
y
x
R θ
! = arc length
! =!R
! = R!
R,!
R
Axis of rotation
θ in radians!
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 17
12 θθθ −=Δ
R
Axis of rotation Axis of rotation
θΔ
Angular Displacement
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 17
12
12
ttt −
−=
Δ
Δ=
θθθω
! = lim!t" 0
!"!t
=d"dt
dtd
ttωω
α =Δ
Δ→Δ
=0
lim12
12
ttt −
−=
Δ
Δ=
ωωωα
Angular Velocity & Acceleration Average angular velocity
Instantaneous angular velocity
Average angular acceleration
Instantaneous angular acceleration
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 17
Clicker Quiz
Two children are on a merry-go-round which is turning with a period of rotation of 20 s. If child A is 3 m and child B is 5 m from the axis of rotation, what is the difference in their angular velocities?
A) 0 rad/s B) π/10 rad/s C) π/20 rad/s D) 2π rad/s
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 17
An object at rest begins to rotate with a constant angular acceleration. If this object rotates through an angle θ in the time t, through what angle did it rotate in the time t/2 ?
A) θ B) θ/2
C) θ/4
D) 2θ
E) 4θ
12
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 17
Clicker Quiz
Angular vs Linear • Angular variables need an axis to be defined • Each point on a rotating rigid body has the same
angular displacement, velocity, and acceleration! • The corresponding linear (or tangential) variables
depend on the radius
vtan =d!dt d! = Rd!! = R! vtan = R
d!dt
atan =dvtandt
= R!
= R d!dt
= R!
!atotal =!atan +
!aRaR =vtan2
R=! 2R
!aR
!atan
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 17
Frequency and Period We can relate the angular velocity of rotation to the frequency of rotation:
1 rev/s =2! rad/s
f = 1T=!2"
1 hertz (Hz) = 1 rev/s
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 17
Vector Nature of Angular Quantities Both ω and α can be treated as vectors
– Choose vector in direction of axis of rotation – But which direction?
Right Hand Rule -Curl fingers on right hand to trace rotation of object -Direction of thumb is vector direction for angular velocity, acceleration -z
+z
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 17
Rotation vs. Translation
The equations of motion for translational and rotational motion (for constant acceleration) are identical
! =!o +"ot +12#t2
2oωω
ω+
=2
)(221
22
2
o
oo
oo
o
vvv
xxavv
attvxx
atvv
+=
−+=
++=
+=
( )oo θθαωω −+= 222
! =!o +"t
!! v
!! a
!! x
t! t
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 17
F
F
F F
F
F
No net force
No motion
rotation
F
F
F translation
translation
translation and rotation
Net force
Motion of extended objects
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 17
The diagram shows the top view of a door, hinge to the left and door-knob to the right. The same force F is applied differently to the door. In which case is the turning ability provided by the applied force about the rotation axis greatest?
Clicker Quiz
A B C
D
E
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 17
Torque : turning force
Force Lever arm Angle
F
axis θ r
r
F θ
What causes rotation? F
F
Rotational Dynamics
F
θ
Fsinθ
Fcosθ ! = rF sin"
F
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 17
Torque : turning force
axis r
Torque causes angular acceleration In analogy with F = ma
Rotational Dynamics
F
θ
Fsinθ
Fcosθ
! = I"
! = rF sin"
I is the rotational equivalent of mass Moment of Inertia or Rotational Inertia
! = r(F sin" )
! = F(rsin" )
θ
r sinθ
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 17
Summary
Rotational Motion Rotational kinematics Rotational dynamics Torque
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 17
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