1. Centre of Mass Definition Total momentum of a system of particles
Motion of the centre of mass
Serway and Jewett
Physics 1D03 - Lecture 16

2. Review: Newton’s Second Law
For a particle:
(Net external force) = ma
For a particle or a system of particles:
(Net external force) = F = dp/dt
(Net external impulse) = I = Dp
Physics 1D03 - Lecture 16

3. Apply Newton’s Laws to objects that are not particles:
e.g., F F or
How will an extended body move (accelerate) when a force is
applied at an arbitrary location?
The motion of the centre of mass is simple; in addition, various parts of the object move around the centre of mass.
Physics 1D03 - Lecture 16

4. Centre of Mass rCM Recall: Definition: m1 CM x or, m2 m3
For continuous objects,
(Recall the position vector r has components x, y, z.)
Physics 1D03 - Lecture 16

5. Extra examples with integrals – non-uniform stick, linear density λ=λo and length L.
Physics 1D03 - Lecture 16

6. Extra examples with integrals – uniform solid semisphere
Physics 1D03 - Lecture 16

7. Dynamics of a system of particles
CM definition:
Differentiate with respect to time:
The total momentum of any collection of particles is equal to the total mass times the velocity of the CM point—amazing but true!
So Newton’s second law gives (differentiate above):
This is remarkable. The motion of the centre of mass is the same as
if the object were a single particle at the CM, with all external forces applied directly to it.
Physics 1D03 - Lecture 16

8. Example v0 x CM
A neutron (mass m) travels at speed v0 towards a stationary deuteron (mass 2m). What is the initial velocity of the CM of the system (neutron plus deuteron)?
Since:
Physics 1D03 - Lecture 16

9. 1/3v0 v0 CM
1/3v0
Physics 1D03 - Lecture 16

10. Examples:
A springboard diver does a triple reverse dive with one and a half twists. Her CM follows a smooth parabola (external force is gravity).
A paddler in a stationary canoe (floating on the water, no friction) walks from the rear seat to the front seat. The CM of the canoe plus paddler moves relative to the canoe, but not relative to the land (the canoe moves backwards). Here there is no external force.
Pendulum cart
Physics 1D03 - Lecture 16

11. MOON ,000 km
Quiz F
A space station consists of a 1000-kg sphere and a 4000-kg sphere joined by a light cylinder. A rocket is fired briefly to provide a 100-N force for 10 seconds. Compare the velocity (CM) change if the rocket motor is
A) at the small sphere
B) at the large sphere C) same for either F
In which case will the space station get to the moon faster?
Physics 1D03 - Lecture 16

12. MOON ,000 km
Quiz F
A space station consists of a 1000-kg sphere and a 4000-kg sphere joined by a light cylinder. A rocket is fired briefly to provide a 100-N force for 10 seconds. In which case will the space station rotate faster?
A) at the small sphere
B) at the large sphere C) same for either F
Physics 1D03 - Lecture 16

13. Still more amazing CM theorems:
1) Kinetic Energy = ½ Mv 2CM + (K.E. relative to CM)
(e.g., rigid body: K = ½ Mv 2CM + ½ ICM w 2 )
2) (Torque about CM) = ICM a , even for an accelerated body
“alpha”
(angular acceleration)
Physics 1D03 - Lecture 16

14. Summary ptotal = M vCM (net external force) = M aCM
Physics 1D03 - Lecture 16