Physics 111: Elementary Mechanics – Lecture 11 Carsten Denker NJIT Physics Department Center for Solar–Terrestrial Research
Physics 111: Elementary Mechanics – Lecture 11 Carsten Denker NJIT Physics Department Center for Solar–Terrestrial Research.
Dec 19, 2015
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Physics 111: Elementary Mechanics – Lecture 11
Carsten Denker
NJIT Physics DepartmentCenter for Solar–Terrestrial
Research
November 14, 2006 Center for Solar-Terrestrial Research
Rolling
Smooth rolling motion
Rotation and translation Reference frame
comv R
November 14, 2006 Center for Solar-Terrestrial Research
Kinetic Energy
Stationary observer
Parallel–axis theorem
A rolling object has two types of kinetic energy: a rotational kinetic energy due to its rotation about its center of mass and a translational kinetic energy due to translation of its center of mass.
2
2com
2 2 2com
2 2com com
1
21
21
2
1 1
2 2
P
P
com
K I
I I MR
K I MR
v R
K I Mv
November 14, 2006 Center for Solar-Terrestrial Research
Forces
Yo–Yo
A net force acting on a rolling wheel speeds it up or slows it down and causes an acceleration.
The acceleration tends to make the wheel slide.
A static frictional force acts on the wheel to oppose that tendency.
netF
sf
November 14, 2006 Center for Solar-Terrestrial Research
Torque
Definition of Torque
Vector (cross) product
(Right–hand rule, order does matter!)
Revisit Chapter III!
r F
November 14, 2006 Center for Solar-Terrestrial Research
Angular Momentum
System of particles
Angular counterpart of linear momentum!
Definition
[kg m2/s] l r p m r v
1 21
n
n ii
L l l l l
November 14, 2006 Center for Solar-Terrestrial Research
Newton’s 2nd LawSingle particle
Linear form
Single particle
Angular form
Proof
The (vector) sum of all torques acting on a particle is equal to the time rate of change of angular momentum of that particle!
net
dpF
dt
net
dl
dt
net
net
l m r v
dl dv drm r v
dt dt dt
m r a v v
r ma r F
r F
November 14, 2006 Center for Solar-Terrestrial Research
Conservation of Angular Momentum
Angular momentum of a solid body about a fixed axis
Law of conservation of angular momentum
(Valid from microscopic to macroscopic scales!)
If the net external torque acting on a system is zero, the angular momentum of the system remains constant, no matter what changes take place within the system
L I
const. i fL L L
net
L
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