Lecture 8 - Center for Solar-Terrestrial Research
Download
Report
Transcript Lecture 8 - Center for Solar-Terrestrial Research
Physics 111: Elementary
Mechanics – Lecture 8
Carsten Denker
NJIT Physics Department
Center for Solar–Terrestrial Research
Introduction
Collisions
Impulse and Linear Momentum
Single Collision
Series of Collisions
Momentum and Kinetic Energy
Inelastic Collisions in One Dimension
One-Dimensional Collision
Completely Inelastic Collision
Elastic Collisions in One Dimension
Collisions in Two Dimensions
October 24, 2006
Center for Solar-Terrestrial Research
Center of Mass for a System of Particles
The center of mass of a body or a system of bodies moves
as though all of the mass were concentrated there and all
external forces were applied there.
xcom
m1 x1 m2 x2
m1 m2
2 bodies, 1 dimension
n bodies, 3 dimensions
1 n
1
xcom
mi xi , ycom
M i 1
M
rcom
1
M
n
m r
i 1
October 24, 2006
i i
n
m y
i 1
i
i
and zcom
1
M
n
m z
i 1
i i
n bodies, 3 dimensions, vector equation
Center for Solar-Terrestrial Research
Center of Mass for a Solid Body
xcom
1
M
xdm, ycom
1
M
ydm and zcom
1
M
zdm
Differential mass element dm
Uniform density
dm M
dV V
xcom
1
1
1
xdV , ycom ydV and zcom zdV
V
V
V
October 24, 2006
Center for Solar-Terrestrial Research
Newton’s 2nd Law for a
System of Particles
System of particles
Fnet Macom
A firework rocket explodes
A grand jeté
October 24, 2006
Center for Solar-Terrestrial Research
Linear Momentum
Particle
p mv
System
dp
Fnet
dt
P Mvcom
dP
Fnet
dt
Conservation of Linear Momentum
P const. Pi Pf
If no net external force acts on a system of particles, the total linear
momentum P of the system cannot change.
If the component of the net external force on a closed system is zero
along an axis, then the component of the linear momentum along that
axis cannot change.
October 24, 2006
Center for Solar-Terrestrial Research
Impulse and Linear Momentum
Definition of Impulse
dp F t dt
pf
pi
Collision of two
particle-like bodies
dp F t dt
tf
ti
J F t dt
tf
ti
Impulse–Linear
Momentum Theorem
p f pi p J
Steady stream of projectiles
October 24, 2006
Center for Solar-Terrestrial Research
Momentum and Kinetic Energy
Closed system (no mass enters or leaves)
Isolated system (no external net force)
Elastic collision (kinetic energy conserved)
Inelastic collision (kinetic energy not conserved)
Completely inelastic collision (bodies always stick
together)
In a closed, isolated system containing a collision, the linear
momentum of each colliding body may change but the total
momentum P of the system cannot change, whether the
collision is elastic or inelastic.
October 24, 2006
Center for Solar-Terrestrial Research
Inelastic Collisions in 1D
Conservation of Linear Momentum
p1i p2 f p1 f p2 f
Completely Inelastic Collision
m1v1i m1 m2 v
m1
v
vi
m1 m2
Velocity of Center of Mass
P Mvcom m1 m2 vcom
vcom
p p2 i
P
1i
m1 m2 m1 m2
October 24, 2006
Center for Solar-Terrestrial Research
Elastic Collisions in 1D
In an elastic collision, the kinetic energy of each colliding body may
change, but the total kinetic energy of the system does not change.
Stationary Target
m1v1i m1v1 f m2 v2 f
1
1
1
m1v12i m1v12f m2 v22 f
2
2
2
m m2
v1 f 1
v1i
m1 m2
v2 f
2m1
v1i
m1 m2
October 24, 2006
Moving Target
m1v1i m2v2i m1v1 f m2v2 f
1
1
1
1
m1v12i m2 v22i m1v12f m2 v22 f
2
2
2
2
m m2
2m2
v1 f 1
v1i
v2i
m1 m2
m1 m2
v2 f
2m1
m m1
v1i 2
v2i
m1 m2
m1 m2
Center for Solar-Terrestrial Research