1. Chapter 9a - Systems of Particles
Center of Mass
point masses
solid objects
Newton’s Second Law for a System of Particles
Linear Momentum for a System of Particles
Conservation of Linear Momentum
Rockets
Internal Energy/External Forces

2. Calculating the center of mass – point objects – 1 D

3. Calculating the center of mass – point objects – 2 D

4. Problem 1
Three masses located in the x-y plane have the following coordinates:
2 kg at (3,-2)
3 kg at (-2,4)
1 kg at (2,2)
Find the location of the center of mass

5. Calculating the center of mass – solid objects – 1 D

6. Calculating the center of mass – solid objects – 2 D

7. Finding the COM

8. Problem 2
What is the center of mass of the Letter “F” shown if it has uniform density and thickness?
2cm
20cm
2cm
5cm
10 cm
2cm
15cm

9. Problem 3 The blue disk has a radius 2R
The white area is a hole in the
Disk with radius R.
Where is the center of mass?

10. COM and translational motion
First time derivative
COM Momentum
Second time derivative
Newton’s 2nd Law

11. What this means….
The sum of all forces acting on the system is equal to the total mass of the system times the acceleration of the center of mass.
The center of mass of a system of particles with total mass M moves like a single particle of mass M acted upon by the same net external force.

12. Conservation of Linear Momentum
If 2 (or more) particles of masses m1, m2, … form an isolated system (zero net external force), then total momentum of the system is conserved regardless of the nature of the force between them.

13. Problem 1
An astronaut finds himself at rest in space after breaking his lifeline. With only a space tool in his hand, how can he get back to his ship which is only 10 m away and out of his reach.

14. Variable mass – Rocket propulsion
small

15. Rocket thrust Find: Thrust, initial net force,
net force as all fuel expended