Transcript Systems of Particles

Systems of Particles
Rigid Bodies
• Rigid Bodies - A collection of particles that
do not move relative to each other.
• What forces are present with the 4 blocks
given below?
m4
m2
m3
m1
Center of Mass
2

d ri
Ftotal   Fi   mi 2
dt
i
i



2
2
2
d mi ri d  mi ri
d   mi ri 
Ftotal  

M 2
2
2

dt
dt
dt
M
i


Total external force dictates the
motion of the center of mass of
the collection of objects.

R

 mi ri
M
Earth & Moon
• Where is the center of mass between the
earth and the moon?
M e  5.97 10 kg
24
mm  7.35 10 kg
22
rem  3.85  10 m
8
rem
Balancing an ‘E’
• Where is the center of mass for the metal
plate illustrated below?
(5 cm, 3 cm)
(0 cm, 0 cm)
Momentum
• Momentum - Newton called it “Quantity of
motion.”


p  mv
• Ex. Find the momentum of a 40 kg ball

2 ˆ
traveling with velocity v   5.0t i  6.0t ˆjm / s
Momentum
• Total momentumfor a system of particles

P   pi
• Newton’s law for multiple
 particles

becomes
dP
Fnet 
dt
• If there is no net external force then
dP
0
dt

P  const.
Exploding Shells
• A cannon shell is fired at 100 m/s with an
angle of 60° w.r.t. horizontal. At the top of
the trajectory, it explodes into two pieces of
equal mass. The one piece has no velocity
immediately after the explosion. What is
the velocity of the second piece and where
do the pieces land?
Kinetic Energy
• For multiple particles
K  mv
1
2
2
i i
  ~
• Velocity w.r.t. to center of mass vi  V  vi
• Then,



 ~  ~
K   12 mi V  vi  V  vi


2
2
~
~
1
1
K   2 miV   miV  vi   2 mi vi
K  K cm  0  Kint
Exploding Shell
• A 10 kg shell is traveling with a velocity of
40iˆm / s. It explodes into two pieces, one
which is 3 kg traveling at 50iˆ  10 ˆjm / s .
What is the velocity of the 2nd piece and
the energy released in the explosion?
Conservation Laws
• Conservation of Momentum
– Valid when collision takes place fast enough.
– External forces can’t change momentum
significantly.
• Conservation of Energy
– Elastic Collision - Kinetic energy is conserved.
– Inelastic Collision - Kinetic energy is not
conserved. (Totally inelastic - Objects stick)
Identify Collisions
• Classify the following collisions
–
–
–
–
–
–
–
Two billiard balls collide.
Two cars collide and lock bumpers.
A bat strikes a baseball.
A ball of clay is thrown at a wall and sticks.
Two nitrogen molecules in the air collide.
A receiver leaps and catches a football.
Two hydrogen atoms in the air collide.
Flying Tackle
• An 85 kg running back is traveling
downfield at 9 m/s. A 120 kg tackle hits
him 30° from head on at a speed of 7 m/s.
Assuming both are not in contact with the
ground during the tackle and they move as
one after the tackle, what is their velocity
after the tackle?
Elastic Collisions
• In 1-D can get 2 equations for 2 unknowns
• Ex. Rutherford studied the composition of
matter by scattering alpha particles off of
thin sheets of gold. If an alpha particle hits
a stationary gold atom head on with a
velocity of 200 m/s, what are the final
velocities of both particles?
Elastic Collisions
(2-D)
• 2-D can get 3 equations for 3 unknowns
• Ex. The cue ball hits a stationary pool ball
with a velocity of 1.0 m/s. If the stationary
ball travels away from the collision at an
angle of 30°, what are the velocities of the
two balls after the collision?
30°
1.0 m/s
Impulse
• From Newton’s 2nd Law

 dp
F
dt


 Fdt   dp
• Impulse - Change in momentum
 t2 

J   Fdt  p
t1
If constant force
or average
force


Ft  p
Bouncing Ball
• A 0.15 kg ball strikes the floor with a
velocity of 15.2iˆ  4.3 ˆjm / s . Its velocity
after striking the floor is 13.0iˆ  3.2 ˆjm / s .
What is the average force the floor exerts on
the ball, if the ball is in contact with the
floor for 0.25s?