### Center of mass

```Physics 7C lecture 09
2D collision, center of
mass
Tuesday October 29, 8:00 AM – 9:20 AM
Engineering Hall 1200
Review: Momentum and Newton’s second law
• The momentum of a
particle is the product
of its mass and its
velocity:
• Newton’s second
law can be written in
terms of momentum
as
Impulse and momentum
• The impulse of a force is the
product of the force and the
time interval during which it
acts.
• On a graph of Fx versus time,
the impulse is equal to the area
under the curve, as shown in
Figure 8.3 to the right.
• Impulse-momentum
theorem: The change in
momentum of a particle during
a time interval is equal to the
impulse of the net force acting
on the particle during that
interval.
Conservation of momentum
• External forces (the normal
force and gravity) act on the
skaters shown in Figure 8.9
at the right, but their vector
sum is zero. Therefore the
total momentum of the
skaters is conserved.
• Conservation of momentum:
If the vector sum of the
external forces on a system
is zero, the total momentum
of the system is constant.
Objects colliding along a straight line
• Two gliders collide on an air track in Example 8.5.
• Follow Example 8.5 using Figure 8.12 as shown below.
Elastic collisions
•In an elastic collision, the
total kinetic energy of the
system is the same after the
collision as before.
Q8.8
An open cart is rolling to the
left on a horizontal surface. A
package slides down a chute
and lands in the cart. Which
quantities have the same value
just before and just after the
package lands in the cart?
A. the horizontal component of total momentum
B. the vertical component of total momentum
C. the total kinetic energy
D. two of A., B., and C.
E. all of A., B., and C.
A8.8
An open cart is rolling to the
left on a horizontal surface. A
package slides down a chute
and lands in the cart. Which
quantities have the same value
just before and just after the
package lands in the cart?
A. the horizontal component of total momentum
B. the vertical component of total momentum
C. the total kinetic energy
D. two of A., B., and C.
E. all of A., B., and C.
A two-dimensional collision
• Two robots collide
and go off at
different angles.
• If the angles are
known, find out
final speeds.
A two-dimensional collision
• momentum
conservation along
x and y.
An automobile collision
• Two cars traveling at right angles collide into each other,
find out the final speed and direction.
An automobile collision
momentum conservation:
A two-dimensional elastic collision
• find out α, β and vb2 after elastic collision.
A two-dimensional elastic collision
Center of mass
• We have talked about the motion
of cars, planets etc. as if they are
a single point. Why can we do
this?
• What is unique about the white
point in the falling wrench?
Center of mass
• The center of mass, of a
distribution of mass in space is
the unique point where the
weighted relative position of the
distributed mass sums to zero
Center of mass
• for a continuous volume
Center of mass of a water molecule
• Where is the center of mass of a water molecule?
Center of mass of a water molecule
• Where is the center of mass of a water molecule?
Q8.9
A yellow block and a red rod are joined together. Each object is
of uniform density. The center of mass of the combined object is
at the position shown by the black “X.”
Which has the greater mass, the yellow block or the red rod?
A. the yellow block
B. the red rod
C. They both have the same mass.
D. not enough information given to decide
A8.9
A yellow block and a red rod are joined together. Each object is
of uniform density. The center of mass of the combined object is
at the position shown by the black “X.”
Which has the greater mass, the yellow block or the red rod?
A. the yellow block
B. the red rod
C. They both have the same mass.
D. not enough information given to decide
Center of mass of symmetrical objects
• It is easy to find the center
of mass of a homogeneous
symmetric object.
Motion of the center of mass
• The total momentum of a system
is equal to the total mass times
the velocity of the center of mass.
• The center of mass of the wrench
moves as though all the mass
were concentrated there.
Tug-of-war on the ice
• When James moved 6 m toward the mug, how far has
Ramon moved?
Tug-of-war on the ice
• Using momentum conservation:
Tug-of-war on the ice
• Using motion of center of mass:
External forces and center-of-mass motion
• When a body or collection of particles is acted upon by
external forces, the center of mass moves as though all the
mass were concentrated there.
External forces and center-of-mass motion
• Fragments of a firework shell would fly at 100 m/s for 5
seconds before they burn out. If a shell reaches its max
height of 1000 meter and explodes, are the audiences on the
ground safe from burning fragments? Ignore air resistance.
External forces and center-of-mass motion
• Fragments of a firework shell would fly at 100 m/s for 5 seconds before they
burn out. If a shell reaches its max height of 1000 meter and explodes, are the
audiences on the ground safe from burning fragments? Ignore air resistance.
motion of center of mass:
motion of fragments relative to center of mass:
Rocket propulsion
• As a rocket burns fuel, its mass decreases, as shown
in Figure below.
• What is the speed of rocket if we know the exhaust
speed vex, burning rate λ=dm/dt and initial mass m0?
Rocket propulsion
•
What is the speed of rocket if we know the
exhaust speed vex, burning rate λ=dm/dt and
initial mass m0?
between time t and t + dt, according to momentum conservation:
(m+dm) v = m (v+dv) + dm (v-vex)
Rocket propulsion
•
What is the speed of rocket if we know the
exhaust speed vex, burning rate λ=dm/dt and
initial mass m0?
between time t and t + dt, according to momentum conservation:
(m+dm) v = m (v+dv) + dm (v-vex)
m dv – v dm + (v-vex) dm = 0
(m0- λ t) dv –vex λ dt = 0
dv - λ vex dt /(m0- λ t)= 0
v + vex ln(m0- λ t) = constant
v = v0 + vex ln (m0/(m0- λ t)) = v0 + vex ln (m0/m)