1. Center of mass 1. Definitions From these definitions follows:
If total external force is zero than center of mass experience zero acceleration

2. d1:d2= m2: m1 2. Some basic properties of the CM
Equal Masses: in the middle
d1:d2= m2: m1
Inverted ratio!
Unequal Masses: closer to heavy one m1 m2 d1 d2
A triangle of equal masses
Connect each corner to
the center of the opposite side
Triangle of unequal masses
Connect each corner to the center
of masses of the two other masses

3. 3. Continuous mass distribution
Some basic properties of the CM
Symmetry:
The CM need not be inside the object!
A system composed of many parts: First condense each part to its CM and then treat each CM as a point like particle
1kg
2kg
C of M

4. (the point where the gravitational force can be considered to act)
Example: The disk shown in figure 1 is uniform and has its CM at the center.
Suppose the disk is cut in half and its pieces arranged as shown in figure 2.
Where is the CM of (2) compared to the CM of (1)?
CM of top half
Higher
Lower
At the same level CM
X CM
CM of bottom half
Fig. 1
Fig. 2
Center of gravity
(the point where the gravitational force can be considered to act)
It is the same as the center of mass as long as the gravitational force does not vary among different parts of the object.
It can be found experimentally by suspending an object from different points.

5. Center of mass (examples)
In (a) the diver’s motion is pure translation.
In (b) it is translation plus rotation.
High jumpers have developed a technique where their CM actually passes under the bar as they go over it. This allows them to clear higher bars.
There is one point that moves in the same path a particle would take if subjected to the same force as the diver. This point is the center of mass.

6. Example: Two people on frictionless roller blades, initially at rest, throw a ball back and forth. After a couple of throws, they are
1. Standing where they were initially.
2. Standing farther away from each other.
3. Standing closer together.
4. Moving away from each other.
5. Moving toward each other.
Trajectory of the CM
When a shell explodes, the CM keeps moving along the parabolic trajectory the shell had before the explosion.

7. Rotational dynamics 1. Torque and golden rule of mechanics r1 r2 F2 r1
r1 /d1 = r2 /d2 d1 r2 F1
W1 =W2
F1 d1 = F2 d2
F1 r1 = F2 r2 F1 F2 r1 r2
F1
F1||
Definition of torque:
Golden rule:

8. 2. Comparing linear and angular characteristics of motion
velocity
force
liner momentum
mass
angular velocity
torque
angular momentum
moment of inertia

9. Example: Moment of inertia of a square of side L made with four identical particles of mass m and four massless rods.
By definition: m
Axis L
Axis L m m m m
Axis L m m m
The moment of inertia depends on the position and orientation of the axis

10. m 1 L 2 3 I1 = m(2L)2 + m(2L)2 = 8mL2 I2 = mL2 + mL2 + mL2 = 3mL2
Example: Three identical balls are connected with three identical, rigid, massless rods. The moments of inertia about axes 1, 2 and 3 are I1, I2 and I3. Which of the following is true?
A. I1 > I2 > I3
B. I1 > I3 > I2
C. I2 > I1 > I3 L m 3 2 1
By definition:
I1 = m(2L)2 + m(2L)2 = 8mL2
I2 = mL2 + mL2 + mL2 = 3mL2
I3 = m(2L)2 = 4mL2

11. A. Solid Al B. Hollow Au C. Both the same
Example: Two spheres have the same radius and equal masses. One is made of solid aluminum and the other is a hollow shell of gold. Which one has the biggest moment of inertia about an axis through its center?
Mass is further
away from the axis
Solid aluminum
Hollow gold
A. Solid Al B. Hollow Au C. Both the same

12. Note: I=kMR2, where k 1

13. Example: Which of these get to the bottom of the ramp first?
(Rolling without slipping)
do racing round things
Energy is conserved and the initial energy (at the top) is mgh.
The final energy (at the bottom) is kinetic energy of translation and rotation.
This means that as smaller is energy of rotation as bigger is energy of translation and as higher is the speed of the object.

14. Noether’s Theorem So far we have learned about three
conserved quantities:
Energy
Momentum
Angular Momentum
Conserved quantities are very important in physics
According to Noether’s Theorem,
for each conserved quantity there exists a symmetry of the laws of physics
which “generates” it.