Download Lectures 23, 24

Physics 218: Mechanics
Instructor: Dr. Tatiana Erukhimova
Lectures 23, 24
Center of Mass

1
rcm 
M

 mi ri
N
i 1
N
M   mi
i 1
xcm
1

M
N
m x
i 1
i i
ycm
1

M
N
m y
i 1
i
i
Find the position of the center of mass of the
system of the sun and Jupiter. (Since Jupiter is
more massive than the rest of the planets put
together, this is essentially the position of the
center of mass of the solar system.) Does the
center of mass lie inside or outside the sun?
Motion of the Center of Mass

1
vcm 
M
vcm x
1

M

1
acm 
M

 mi vi
N
i 1
N
m v
i 1
i ix

 mi ai
N
i 1
vcm y
1

M
N
m v
i 1
i iy


Macm   Fi
N
i 1


Macm   Fi
N
i 1
The center of mass of a system moves as if all of
the mass of the system were concentrated at that
point and as if all of the forces were acting at
that point
There is only the external forces that affect the


motion of the center of mass ( F12   F21 )



Macm   Fi external  Fexternal
N
i 1
Only external forces affect the motion of the
center of mass
Momentum is a vector!


p  mv
Vector equation!
N
N




Ptotal   pi   mi vi  Mvcm
i 1
i 1

1
vcm 
M


Ptotal  Mvcm

 mi vi
N
i 1




dvcm dPtotal
Macm  Fexternal  M

dt
dt

dPtotal 
 Fexternal
dt 

If Fexternal  0,
dPtotal
0
dt

Ptotal  Const
Conservation of Momentum
If there is no external force on a system, then
the total momentum of the system is a constant

Ptotal  Const


P(before)  P(after)
True in X and Y directions separately!
total
x
dP
dt
F
total
x
total
y
dP
dt
F
total
y
Problem Solving
For Conservation of Momentum problems:
1. BEFORE and AFTER
2. Do X and Y Separately
Before
Y
X
After
X
Y
Inelastic collision
A collision in which the total kinetic energy after
the collision is not equal to the kinetic energy
before the collision is called an inelastic collision.
BEFORE
vA
vB  0
A
B
AFTER
Vafter?
A
B
Perfectly elastic collision
A collision in which the total kinetic energy after
the collision is the same than that before the
collision is called an elastic collision.
vA
A
B
A block of mass m is moving along x axis with
a velocity of V0. It collides with a block of
mass M, initially at rest.
1) What is the change in kinetic energy of the
system of two balls:
a) if the collision is perfectly elastic;
b) if the collision is perfectly inelastic
(balls stick together after collision).
2) For m = M = m0, find the velocity of each
ball after a perfectly elastic collision.