System of particles - Center of mass - C.o.M Theorem

1. PARTICLES SYSTEMS
CHAPTER 9

3. LIST OF TERMS
• DEFORMABLE SYSTEM
• RIGID SYSTEM
• INTERNAL FORCES
• EXTERNAL FORCES
• CENTER OF MASS
• CENTER OF GRAVITY
• THEOREM OF THE CENTER
OF MASS

13. PARTICLES AND SYSTEMS
• WHEN THE DISTANCE TRAVELED BY AN OBJECT IS
VERY LARGE COMPARED TO ITS DIMENSIONS, THIS
OBJECT CAN BE CONSIDERED AS A PARTICLE.
• A GROUP OF PARTICLES IS CALLED SYSTEM OF
PARTICLES.

14. DEFORMABLE SYSTEM VS. RIGID SYSTEM
System
Deformable: the distance between its points varies.
Non-deformable: the distance between any two points
points remains fixed.

15. MASS CENTER OF A PARTICLES SYSTEM - 2
DIMENSIONS
• SUPPOSE A SYSTEM (S) CONSISTS OF SEVERAL PARTICLES,
P1, P2, P3, …, P 𝑛, OF RESPECTIVE MASSES
𝑚1, 𝑚2, 𝑚3, … , 𝑚 𝑛.
• THEN THE POSITION OF THE CENTER OF MASS '' G '' OF
THE SYSTEM WITH RESPECT TO AN ORIGIN '' O '' IS
GIVEN BY :
𝑥 𝐺 =
𝑚1 ∙ 𝑥1 + 𝑚2 ∙ 𝑥2 + 𝑚3 ∙ 𝑥3 + ⋯ + 𝑚 𝑛 ∙ 𝑥 𝑛
𝑚1 + 𝑚2 + 𝑚3 + … + 𝑚 𝑛
𝑦 𝐺 =
𝑚1 ∙ 𝑦1 + 𝑚2 ∙ 𝑦2 + 𝑚3 ∙ 𝑦3 + ⋯ + 𝑚 𝑛 ∙ 𝑦𝑛
𝑚1 + 𝑚2 + 𝑚3 + … + 𝑚 𝑛

16. 𝒙 𝑮 =
𝟏
𝑴
𝒊=𝟏
𝒊=𝒏
𝒎𝒊 ∙ 𝒙𝒊
𝒚 𝑮 =
𝟏
𝑴
𝒊=𝟏
𝒊=𝒏
𝒎𝒊 ∙ 𝒚𝒊
Where: 𝑴 = 𝒎 𝟏 + 𝒎 𝟐 + 𝒎 𝟑 + … + 𝒎 𝒏

21. G1
G2
(L/2 ; 0)
(0 ; L/2)
L
G ( ? ; ?)

26. WHAT ABOUT DETERMINING THE CENTER OF MASS
FOR A REAL OBJECT?
Figure
5: Plumb line
method being
used to find
the center of
mass of an
irregular
object.

27. Figure 6: Topple
limit of a poorly
loaded truck.

28. Figure 6: Topple
limit of a poorly
loaded truck.

29. Figure 6: Topple
limit of a poorly
loaded truck.

30. Forces
internal: result from interactions between different parts
of the same system.
External: result from interactions between the system or
part of the system and the environment (outside world).
INTERNAL FORCES & EXTERNAL FORCES

32. CENTER OF GRAVITY
• THE POINT OF APPLICATION OF THE
WEIGHT OF A SYSTEM IS CALLED
THE CENTER OF GRAVITY OF THE
SYSTEM.
• FREQUENTLY THE CENTER OF
GRAVITY COINCIDES WITH THE
CENTER OF MASS OF THE SYSTEM.

33. MOTION OF CENTER OF MASS - THEOREM OF
CENTER OF MASS
TO UNDERSTAND ANALYTICALLY THE MOVEMENT OF THE CENTER OF
MASS LET US TAKE AGAIN THE DEFINITION OF THE POSITION VECTOR
OF THE CENTER OF MASS:
𝑟𝐺 =
𝑚1 ∙ 𝑟1 + 𝑚2 ∙ 𝑟2 + 𝑚3 ∙ 𝑟3 + ⋯ + 𝑚 𝑛 ∙ 𝑟𝑛
𝑚1 + 𝑚2 + 𝑚3 + … + 𝑚 𝑛
(𝑚1 + 𝑚2 + 𝑚3 + … + 𝑚 𝑛) ∙ 𝑟𝐺 = 𝑚1 ∙ 𝑟1 + 𝑚2 ∙ 𝑟2 + 𝑚3 ∙ 𝑟3 + ⋯ + 𝑚 𝑛 ∙ 𝑟𝑛
LET'S DERIVE RELATIVE TO TIME
(𝑚1 + 𝑚2 + 𝑚3 + … + 𝑚 𝑛) ∙
𝑑 𝑟𝐺
𝑑𝑡
= 𝑚1 ∙
𝑑 𝑟1
𝑑𝑡
+ 𝑚2 ∙
𝑑 𝑟2
𝑑𝑡
+ 𝑚3 ∙
𝑑 𝑟3
𝑑𝑡
+ ⋯ + 𝑚 𝑛 ∙
𝑑 𝑟𝑛
𝑑𝑡
⇒ 𝑀 ∙ 𝑉𝐺 = 𝑚1 ∙ 𝑉1 + 𝑚2 ∙ 𝑉2 + 𝑚3 ∙ 𝑉3 + ⋯ + 𝑚 𝑛 ∙ 𝑉𝑛
OR

34. MOTION OF CENTER OF MASS - THEOREM OF
CENTER OF MASS
LET'S DERIVE AGAIN RELATIVE TO TIME, WE GET THEN:
THIS RELATION EXPRESSES THE THEOREM OF THE CENTER OF MASS OF A MATERIAL
SYSTEM
⇒
𝑀 ∙
𝑑𝑉𝐺
𝑑𝑡
= 𝑚1 ∙
𝑑 𝑉1
𝑑𝑡
+ 𝑚2 ∙
𝑑 𝑉2
𝑑𝑡
+ 𝑚3 ∙
𝑑 𝑉3
𝑑𝑡
+ ⋯ + 𝑚 𝑛 ∙
𝑑 𝑉𝑛
𝑑𝑡
𝑀 ∙ 𝑉𝐺 = 𝑚1 ∙ 𝑉1 + 𝑚2 ∙ 𝑉2 + 𝑚3 ∙ 𝑉3 + ⋯ + 𝑚 𝑛 ∙ 𝑉𝑛
⇒ 𝑀 ∙ 𝑎 𝐺 = 𝑚1 ∙ 𝑎1 + 𝑚2 ∙ 𝑎2 + 𝑚3 ∙ 𝑎3 + ⋯ + 𝑚 𝑛 ∙ 𝑎 𝑛
𝒇 𝟏 𝒇 𝟐 𝒇 𝟑 𝒇 𝒏

35. MOTION OF CENTER OF MASS - THEOREM OF
CENTER OF MASS
THEOREM OF CENTER OF MASS:
THE MOVEMENT OF THE CENTER OF MASS OF A SYSTEM IS THAT OF A
PARTICLE, OF MASS EQUAL TO THAT OF THE SYSTEM, AND
SUBJECTED TO A FORCE EQUAL TO THE SUM OF THE EXTERNAL
FORCES APPLIED TO THE SYSTEM.

38. MOTION OF CENTER OF MASS - THEOREM OF
CENTER OF MASS
PARTICULAR CASE
IF THE SUM OF THE EXTERNAL FORCES IS ZERO ( ), THE
CENTER OF MASS OF THE SYSTEM CAN BE EITHER AT REST OR IN
UNIFORM RECTILINEAR MOTION. INDEED :
IF 𝑀 ∙ 𝑎 𝐺 = 0 ⇒ 𝑎 𝐺 = 0 ⇒ 𝑉𝐺 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
⇒
"𝐺" 𝑟𝑒𝑚𝑎𝑖𝑛𝑠 𝑎𝑡 𝑟𝑒𝑠𝑡 𝑖𝑓 𝑖𝑡 𝑖𝑠 𝑖𝑛𝑖𝑡𝑖𝑎𝑙𝑙𝑦 𝑎𝑡 𝑟𝑒𝑠𝑡
𝑜𝑟
"𝐺" 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑒𝑠 𝑖𝑛 𝑈𝑅𝑀 𝑖𝑓 𝑖𝑡 𝑖𝑠 𝑖𝑛𝑖𝑡𝑖𝑎𝑙𝑙𝑦 𝑚𝑜𝑣𝑖𝑛𝑔