Transcript Only external forces affect the motion of the center of mass

Physics 218: Mechanics
Instructor: Dr. Tatiana Erukhimova
Lectures 25-29
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
Problem 5 from handout
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.
Problem 4 p.200
In a nuclear collision an incoming proton has
initial velocity of magnitude 3.5 105 m/s. It collides
with another proton, initially at rest. After the
collision one proton goes off at 370 to the x axis.
If the collision is perfectly elastic, find the
velocities of the two protons after the collision.
The ballistic pendulum
“Famous Problem” from the book
Problem 5 p. 200
v0
A cannon is mounted on top of a narrow wall:
There is no friction between the wall and the
cannon. The cannon fires a cannon ball with a
horizontal velocity v0=200 m/s. The cannon has
mass 100 kg and the ball mass 10 kg. The height
of the wall is 9.8 m. Find the final positions of the
cannon and the cannon ball.
Quiz
A block of mass m is sliding on a frictionless
table with velocity v0. It explodes into two
pieces, one with mass m/3. The light piece flies
off horizontally, perpendicular to the original
direction of motion, with velocity 2v0. Find as
many equations as you need to find the velocity
of the heavy piece.
Quiz
You are standing on a frictionless surface. Some
idiot throws a rock at you which you catch. In
terms of your mass, the rock’s mass and the rock’s
velocity find your position as a function of time
after you catch the rock.
Impulse
 t2 


J   Fdt  p2  p1
t1
 dp
F
dt
Changes in a particle’s momentum are due to
impulse, which depends on the time over which
the net force acts.
Impulse
Suppose you throw a ball with a mass of 0.4 kg
against a brick wall. It hits the wall moving
horizontally to the left at 30 m/s and rebounds
horizontally to the right at 20 m/s. a) Find the
impulse of the net force on the ball during its
collision with the wall. b) If the ball is in contact
with the wall for 0.010 s, find the average
horizontal force that the wall exerts on the ball
during the impact.
Quiz
A small car weighing m1 is traveling due north
when it collides with a pick-up truck weighting
m2 which was traveling due east. After the
collision the two vehicles move off together at an
angle θ north of east. The driver of the car
claimed that the truck driver was at fault
because he was exceeding the speed limit, going
with a velocity v1. If this were true, what was the
car’s initial velocity?

Polar coordinates
Have a great day!
Hw: Chapter 12 problems and
exercises
Reading: Chapters 12, 13