CENTER OF
MASS (CM)
A POINT IN THE SYSTEM IN WHICH THE
TOTAL MASS IS CONCENTRATED
The CM of an object or a system of objects may be used to represent the
motion and interactions of the object or the system of objects.
How can the location of the CM of two or more masses be determined?
To determine the location center of mass of two or more masses
lying in an axis and whose location is relative to origin:
To compute for the motion of the center of mass:
HOW WOULD YOU
FIND THE CENTER OF
MASS OF AN
AMMONIA
MOLECULE?
The center of mass is considered as the mass-weighted average position of
the particles or elements making up the system.
EXAMPLE 1
EXAMPLE 2
Three 1.2 kg point particles are placed in the x-y coordinate system, as shown
in the figure. Find the center of mass of this system.
EXAMPLE 3
Four squares, with equal side length of 1.0m, are arranged based on a
rectangular coordinate as shown in the figure. The masses of A, B, C and D
are 1.0 kg, 2.0 kg, 3.0kg, and 4.0 kg, respectively. Find the center of mass of
the system of four squares with respect to the x-and y- axes shown.
EXAMPLE 4
Particle A of mass 2.0 kg is moving at 15.0 m/s to the east. Particle B of mass
3.0kg is also moving to the east at 10.0 m/s. Find the velocity of the center of
the two particles.
EXERCISE
• 1. Find the center of mass of four 2kg bodies placed at each corner of a
square of side 1.5m.
• 2. Find the center of mass of four squares in Example 3 when another square
of side 1.0m and mass 5.0 kg is placed on top of square C.
• 3. Find the velocity of the center of mass of the two particles, particle A of
mass 3.0 kg is moving at 17m/s to the east and particle B of mass 2.O kg is
moving toward west at 12m/s.
MOMENTUM ( LINEAR MOMENTUM)
"mass in motion" vector quantity
An object has a large momentum if both its mass and its velocity
are large.
The momentum of any object that is at rest is 0.
EXAMPLE 1
1. Determine the momentum of a ...
a. 60-kg halfback moving eastward at 9 m/s.
b. 1000-kg car moving northward at 20 m/s.
c. 40-kg freshman moving southward at 2 m/s.
EXAMPLE 2
• A hypervelocity bullet used in a 0.22 long rifle usually weigh around 2.1 g
and can have a speed of 550 m/s. What must be the speed of a 75 kg man to
match the momentum of this bullet?
Thus, it would require a greater amount of force or a longer
amount of time or both to bring such an object to a halt. As the
force acts upon the object for a given amount of time, the
object's velocity is changed; and hence, the object's momentum
is changed.
Impulse is a certain amount of force you apply for a certain amount of time
to cause a change in momentum.
Impulse-momentum
equation
EXAMPLE
Which has the greatest velocity change?
Which has the greatest acceleration?
Which has the greatest momentum change?
Which has the greatest Impulse?
EXAMPLE
•You throw a ball with a mass of 0.40 kg against a brick wall. It
hits the wall moving horizontally to the left at 30m/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?
EXAMPLE
• A neophyte player catches a 125 g ball moving at 25.0 m/s in 0.02 s. A
professional player catches the same ball in 1.0s by slightly
retracting his hand during the catch. Find the forces exerted by the
ball on the hands of the two players?
EXERCISE
• 1. A 2500 kg delivery truck contains 125 cavans of rice,
each of mass 50 kg. It is moving at a speed of 5.0 m/s.
How many cavans of rice must be unloaded at a
nearby grocery so that the truck can moves at 9.0 m/s
and yet maintain the same momentum?
• A soccer ball has a mass of 0.40 kg. Initially it is
moving to the left at 20 m/s but then it is kicked. After
the kick it is moving at 45° upward and to the right
with speed 30 m/s. Find the impulse of the net force
and the average net force, assuming a collision time
Δt= 0.010s.
CONSERVATION OF LINEAR MOMENTUM
❑Internal forces- are forces
that the particles of a For an isolated system, the
system exert on one another.
❑External forces- forces
total momentum before
exerted on any part of the interaction equals the total
system by other objects momentum after interaction
outside the system.
* If there are no external forces 𝑚𝐴𝑣𝐴1 + 𝑚𝐵𝑣𝐵1 = 𝑚𝐴𝑣𝐴2 + 𝑚𝐵𝑣𝐵2
acting on a system, then the
system is said to be isolated.
CONSERVATION OF MOMENTUM
▪ When a person walks on a
skateboard in one direction,
the skateboard moves in the
opposite direction.
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CONSERVATION OF MOMENTUM
▪ When a person steps from a
small boat on to a dock. As
the person steps toward the
dock, the boat moves away
from the dock.
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CONSERVATION OF MOMENTUM
▪ Conservation of momentum is
also applied in rocket propulsion.
Once fired, the exhaust gases
from the fuel shoot down at great
speed. The downward
momentum of these gases is
balanced by the upward
momentum of the rocket. The
upward force that the exhaust
gases exert on the rocket is
called the thrust.
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CONSERVATION OF MOMENTUM
▪ Using the impulse-momentum theorem, assuming that the
speed v at which the exhaust gases ejected is constant, the
thrust F can be obtained by;
∆𝑚
F= v ∆𝑡 where
∆m
∆t
- the rate at which fuel is consumed
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EXAMPLE 1
• A 50.0kg student uses an improvised 75kg raft to cross a heavily
flooded street. He noticed that as he jumps to the sidewalk opposite
the street with a speed of 1.5m/s relative to the flood, the raft moves
away. With what speed will the raft move relative to the flood?
Assume that the raft is stationary before the student jumps to the
sidewalk.
EXAMPLE 2
•A 100 Kg footballer, running at 5m/s collides with a 70
kg defending footballer (running at 4m/s) as they get
tackled. If the defender holds on to footballer, what
will be their final velocity after they collide?
EXAMPLE 3
•Two gliders with different masses
move toward each other on a
frictionless air track . After they
collide, glider B has a final
velocity. What is the final velocity
of glider A? How do the changes
in momentum and in velocity
compare?
CHECK YOUR UNDERSTANDING
A marksman holds a rifle of mass loosely, so it can recoil
freely. He fires a bullet of mass horizontally with a velocity
relative to the ground of what is the recoil velocity of the
rifle? What are the final momentum and kinetic energy of the
bullet and rifle?
COLLISIONS
•IS AN INTERACTION BETWEEN TWO OR MORE BODIES
THAT COME IN CONTACT WITH EACH OTHER.
•MOMENTUM IS ALWAYS CONSERVED DURING
COLLISIONS
COEFFICIENT OF RESTITUTION (e)
• IS THE NEGATIVE RATIO OF THE VELOCITY OF TWO COLLIDING BODIES
AFTER COLLISION TO THE RELATIVE VELOCITY BEFORE COLLISION.
𝑣𝐴2 𝑣𝐵2
e= − −
(𝑣𝐴1 𝑣𝐵1
− )
TYPES OF COLLISION
• Elastic Collision- the bodies separate after colliding each other
• Both momentum and kinetic energy are conserved during collision.
• The coefficient of restitution is equal to 1
• Inelastic Collision
• Momentum is conserved but not the kinetic energy
• Some of the kinetic energy goes into other forms like heat, sound, and permanent deformation of a given
system.
• The coefficient of restitution for inelastic collision is between 0 and 1.
• Completely Inelastic Collision- if the interacting bodies stick together and move as one after collision
Elastic Collision Inelastic Collision
The total kinetic energy is conserved. The total kinetic energy of the bodies at the
beginning and the end of the collision is different.
Momentum is conserved. Momentum is conserved.
No conversion of energy takes place. Kinetic energy is changed into other energy such as
sound or heat energy.
Highly unlikely in the real world as there is almost This is the normal form of collision in the real world.
always a change in energy.
An example of this can be swinging balls or a An example of an inelastic collision can be the
spacecraft flying near a planet but not getting collision of two cars.
affected by its gravity in the end.
EXAMPLE
• 1. Two balls A and B are approaching each other with velocities 4.5
m/s and 7.2 m/s, respectively. The mass of ball A is 3.2 kg, while that
of ball B is 5.6 kg. (a) find the velocity of the two bodies after impact,
assuming that the collision is perfectly inelastic, (b) Find the kinetic
energy lost during collision.
• 2. Suppose the collision in Sample 1 is elastic. Find (a) how much
kinetic energy is lost and (b) how fast each ball is moving after
colliding.
EXERCISE
• Car A of mass 2000kg moving at 72km/h, east tries to overtake car B of
mass 1250 kg moving at 60km/h, east. Car A collides with car B. the cars
stick together after collision. Find their velocity after collision.
• A 2.25 kg ball moving with a speed of 1.75 m/s toward the east strikes a
stationary ball of equal mass. Find the velocities of the ball after an
elastic collision.
