This document discusses the concepts of centroid, center of gravity, momentum, and impulse. It begins by defining the centroid as the geometrical center of an area, while the center of gravity is the point where all the mass of a body can be assumed to be concentrated. It then explains how momentum is calculated as mass times velocity, and how impulse is equal to the change in momentum caused by an applied force over time. Several examples are provided to illustrate these concepts, such as how changing the time over which a force is applied affects the force magnitude. The document aims to build an understanding of these foundational physics concepts.

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RECALL

2. GEOMETRICAL CENTRE AND THE
CENTER OF GRAVITY

4. Centroids
The centroid of an area is situated at its geometrical
centre. In each of the following figures ‘G’ represents the
centroid, and if each area was suspended from this point it
would balance.

5. Center of gravity or mass
The centre of gravity of a body is:
•The point at which all the mass of the body may be
assumed to be concentrated.
•The point through which the force of gravity is
considered to act vertically downwards, with a force
equal to the weight of the body.
•The point about which the body would balance.
The centre of gravity of a homogeneous body is at its
geometrical centre.

6. See-Saws
We all remember the fun
see-saw of our youth.
But what happens if . . .

7. Difference Between Centre of Gravity and Centroid
Centre of Gravity Centroid
The point where the total
weight of the body focuses
upon
It is referred to as the
geometrical centre of a body
It is the point where the
gravitational force (weight)
acts on the body
It is referred to the centre of
gravity of uniform density
objects
It is denoted by g It is denoted by c
Centre of Gravity in a uniform
gravitational field is the
average of all points, weighted
by local density or specific
weight
The centroid is a point in a
plane area in such a way that
the moment of area about any
axis throughout that point is 0
It is a physical behaviour of
the object, a point where all
the weight of an object is
acting
It is a geometrical behaviour.
It is the centre of measure of
the amount of geometry.

8. Balancing Unequal Masses
Moral
Both the masses and their positions
affect whether or not the “see saw”
balances.

9. Balancing Unequal Masses
Need:
M1 d1 = M2 d2
M1
M2
d1 d2

10. Changing our Point of
View
The great Greek mathematician
Archimedes said, “give me a
place to stand and I will move
the Earth,” meaning that if he
had a lever long enough he
could lift the Earth by his own
effort.

11. In other words. . .
We can think of leaving the masses in place and moving the
fulcrum.
It would have to be a pretty
long see-saw in order to
balance the school bus and
the race car, though!

12. In other words. . .
(We still) need:
M1 d1 = M2 d2
M2
d1 d2
M1

13. What happens if there are many
things trying to balance on the
see-saw?
Where do we place the fulcrum?
Mathematical Setting
First we fix an origin and a coordinate system. .
.
0 1
-1
-2 2

14. Mathematical Setting
And place the objects in the coordinate system.
. .
0
M2
M1
M3
M4
d2
d1
d3 d4
Except that now d1, d2, d3, d4, . . . denote the placement of the
objects in the coordinate system, rather than relative to the
fulcrum.
(Because we don’t, as yet, know where the fulcrum will be!)

15. Mathematical Setting
And place the objects in the coordinate system.
. .
0
M2
M1
M3
M4
d2
d1
d3 d4
Place the fulcrum at some coordinate .
is called the center of mass of the system.
x
x
x

16. Mathematical Setting
And place the objects in the coordinate system.
. .
0
M2
M1
M3
M4
d2
d1
d3 d4
In order to balance 2 objects, we needed:
M1 d1 = M2 d2 OR M1 d1 - M2 d2 =0
For a system with n objects we need:
x
1 1 2 2 3 3
( ) ( ) ( ) ( ) 0
n n
M d x M d x M d x M d x
        

17. Finding the Center of Mass of the System
1 1 2 2 3 3
( ) ( ) ( ) ( ) 0
leads to the following set of calculations
n n
M d x M d x M d x M d x
        
x
1 1 1 2 2 2 3 3 3 0
n n n
M d M x M d M x M d M x M d M x
        
Now we solve for .
1 1 2 2 3 3 1 2 3
n n n
M d M d M d M d M x M x M x M x
        
 
1 1 2 2 3 3 1 2 3
n n n
M d M d M d M d M M M M x
        
1 1 2 2 3 3
1 2 3
And finally . . .
n n
n
M d M d M d M d
x
M M M M
   

   

18. The Center of Mass of the System
1 1 2 2 3 3
1 2 3
n n
n
M d M d M d M d
x
M M M M
   

   
In the expression
The numerator is called the
first moment of the system
The denominator is the
total mass of the
system

19. Let’s start with everyday
language
What do you say when a sports
team is on a roll?
They may not have the lead but
they may have ___________
MOMENTUM
A team that has momentum is hard
to stop.

20. Momentum and
Impulse

21. What is Momentum?
An object with a lot of
momentum is also hard to
stop
Momentum = p = mv
Units: kg∙m/s^2
m=mass
v=velocity
Momentum is also a vector (it
has direction)

22. Let’s practice
•
A 1200 kg car drives west at 25 m/s
for 3 hours. What is the car’s
momentum?
•
Identify the variables:
–
1200 kg = mass
–
25m/s, west = velocity
–
3 hours = time
P = mv = 1200 x 25 = 30000 kg m/s^2,
west

23. How hard is it to stop a
moving object?
To stop an object, we have to apply a
force over a period of time.
This is called Impulse
Impulse = FΔt Units: N∙s
F = force (N)
Δt = time elapsed (s)

24. How hard is it to stop a
moving object?
•
Using Newton’s 2nd Law we get
FΔt= mΔv
Which means
Impulse = change in momentum

25. Why does an egg break or
not break?
•
An egg dropped on a tile floor breaks, but
an egg dropped on a pillow does not. Why?
FΔt= mΔv
In both cases, m and Δv are the same.
If Δt goes up, what happens to F, the force?
Right! Force goes down. When dropped on a
pillow, the egg starts to slow down as soon
as it touches it. A pillow increases the
time the egg takes to stops.

26. Practice Problem
A 57 gram tennis ball falls on a tile floor.
The ball changes velocity from -1.2 m/s to
+1.2 m/s in 0.02 s. What is the average
force on the ball?
Identify the variables:
Mass = 57 g = 0.057 kg
Δvelocity = +1.2 – (-1.2) = 2.4 m/s
Time = 0.02 s
using FΔt= mΔv
F x (0.02 s) = (0.057 kg)(2.4 m/s)
F= 6.8 N

27. Car Crash
Would you rather be in a
head on collision with an
identical car, traveling at
the same speed as you,
or a brick wall?
Assume in both situations you
come to a complete stop.
Take a guess
http://techdigestuk.typepad.com/photos/uncategorized/car_crash.J
PG

28. Car Crash (cont.)
Everyone should vote now
Raise one finger if you
think it is better to hit
another car, two if it’s
better to hit a wall and
three if it doesn’t
matter.
And the answer is…..

29. Car Crash (cont.)
The answer is…
It Does Not Matter!
Look at FΔt= mΔv
In both situations, Δt, m, and Δv
are the same! The time it
takes you to stop depends on
your car, m is the mass of
your car, and Δv depends on
how fast you were initially
traveling.

30. Egg Drop connection
•
How are you going to use this in your egg
drop?
Which of these variables can you control?
FΔt= mΔv
Which of them do you want to maximize,
which do you want to minimize
(note: we are looking at the force on the egg.
Therefore, m represents the egg mass, not
the entire mass of the project)