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

Calculating the center of mass – point objects – 1 D

Calculating the center of mass – point objects – 2 D

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

Calculating the center of mass – solid objects – 1 D

Calculating the center of mass – solid objects – 2 D

Finding the COM

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

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?

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

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.

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.

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.

Variable mass – Rocket propulsion small

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