1. Centre of Mass

2. Announcement
All Term1 grades will be finalised tomorrow evening.

3. Goal of the class To understand centre of mass.
To find the centre of mass of irregular shapes.
Question of the day: Which way can the centre of mass be found practically without calculation?

4. Centre of Mass (COM/CM)
Sometimes called the centre of gravity, but centre of mass is more correct.
Buses and sports cars are designed with a very low centre of mass so they don’t tip over as easy.
When you run, you lean forward to shift your centre of mass.

5. Collection of Particlesz y x
Let’s define a coordinate system in x,y and z
And add on a mass m1 at a displacement r1 from the origin. This has coords x1,y1,z1
Do same with m2 and mN
The centre of mass of all the particles will have a single location and will be a point!
CM: (Xcm,Ycm,Zcm)
Xcm = m1x1+m2x2+…..mNxN/m1+m2+…+mN
But m1+m2+…+mN = M
So Xcm = 1/M Sum (mixi)
Same for Y and Z

6. Four Particles Particle 1: m1 = 1 kg (x1,y1,z1) = (1,0,0) m
Draw on particles
Using Xcm = 1/M Sum (mixi) where N=4
M = =6kg
Xcm = 1/6 (15) = 2.5m
Ycm = 0.5m
Zcm = 0m ofcourse
(Xcm, Ycm,Zcm) = (2.5,0.5,0)

7. Extended Objects Divide volume into N small pieces z y x
Draw an irregular shape and divide into small pieces
Each small piece will have a tiny mass and a tiny volume.
Let’s label them dm1, dm2 etc up to dmN
Suppose I pick a piece (the ith piece)
The displacement vector goes to the centre of this piece ri
Masses: dm1,dm2,…dmN have positions (x1,y1,z1),(x2,y2,z2) etc..
We know Xcm is approx 1/M (x1dm1 + x2dm2+…+xNdmN)
= 1/M sum xidmi
But when we take N to be infinite we get the limit N-> inf
Xcm = 1/M integral xdm
Where we integrate over the mass and not over x,y or z
The force he hits the ball with it isn’t constant. In fact if we draw it like so:
When he first hits the ball, at that instant the force is 0. The same when the ball leaves the ball.
The time he hits it over t2-t1 = delta t which will be a few milliseconds only.
J = F_av delta t
Example baseball
Δt = 2x10-3 seconds
What is the average force on the ball
F = Δp/Δt or Δp = FΔt
Find Δp = 12kgm/s
Find F_av = 6x10-3N

8. Centre of Mass of a Rod Draw a rod on the picture
Total length L and mass M
Indicate x->x+dx on pic
We know Xcm = 1/M integral xdm
The mass per unit length is M/L
Therefore dm = M/L dx
Xcm = 1/M integral xdm = 1/M integral x M/L dx = L/2

9. Centre of Mass of a Right Triangle
Draw a rod on the picture
Total length L and mass M
Indicate x->x+dx on pic
We know Xcm = 1/M integral xdm
The mass per unit length is M/L
Therefore dm = M/L dx
Xcm = 1/M integral xdm = 1/M integral x M/L dx = L/2

10. Orbiting Bodies
Virtual centre of mass - doughnut

11. Problem
A uniform rod of mass M and length L is bent into a quarter of a circle. Locate its centre of mass.
Dm mass of length R d(theta) = M/LR d(theta)
But x = Rcos(theta)