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?

m1

m2 m3

m4

Center of Mass

i

ii

iitotal dt

rdmFF

2

2

M

rm

dt

dM

dt

rmd

dt

rmdF iiii

i

iitotal

2

2

2

2

2

2

M

rmR ii

Total external force dictates the motion of the center of mass of the collection of objects.

Earth & Moon• Where is the center of mass between the

earth and the moon?

kgM e241097.5 kgmm

221035.7 mrem

81085.3 rem

Balancing an ‘E’• Where is the center of mass for the metal

plate illustrated below?

(0 cm, 0 cm)

(5 cm, 3 cm)

Momentum• Momentum - Newton called it “Quantity of

motion.”

• Ex. Find the momentum of a 40 kg ball traveling with velocity

vmp

smjtitv / ˆ 0.6ˆ 0.5 2

Momentum• Total momentum for a system of particles

• Newton’s law for multiple particles becomes

• If there is no net external force then

ipP

dt

PdFnet

dt

Pd

0 .constP

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

• Velocity w.r.t. to center of mass

• Then,

221

iivmK

ii vVv ~

iii vVvVmK ~~

21

2212

21 ~~

iiiii vmvVmVmK

int0 KKK cm

Exploding Shell• A 10 kg shell is traveling with a velocity of

. It explodes into two pieces, one which is 3 kg traveling at

. What is the velocity of the 2nd piece and the energy released in the explosion?

smi / ˆ40 smji / ˆ10ˆ50

Impulse• From Newton’s 2nd Law

• Impulse - Change in momentum

dt

pdF

pddtF

pdtFJt

t

2

1

If constant force or average force

ptF

Bouncing Ball• A 0.15 kg ball strikes the floor with a

velocity of . Its velocity after striking the floor is . What is the average force the floor exerts on the ball, if the ball is in contact with the floor for 0.25s?

smji / ˆ3.4ˆ2.15 smji / ˆ2.3ˆ0.13

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