Class Notes, Oct 9

Class Notes for Fall, 2000

Week of	Aug 23	Sep 6	Oct 2	Nov 6	Phy 205 page
	Aug 28	Sep 11	Oct 9	Nov 13
		Sep 18	Oct 16	Nov 20	Notes for Fall, 1999
		Sep 25	Oct 23	Nov 27
			Oct 30

Phy 205, Week 8

9 Oct Briefly recapitulate the derivation of the period of a conical pendulum. Compare theory and experiment. The difference was about 1%.

Show where the concept of center of mass comes from. Look at two masses interacting with each other and also being affected by outside forces. For each mass, I can write the basic Newtonian equation:

F_{on 1} = F_{on 1, by 2} + F_{on 1, external} = m₁ a₁ = m₁ d v₁ / dt = m₁ d² r₁ / dt² F_{on 2} = F_{on 2, by 1} + F_{on 2, external} = m₂ a₂ = m₂ d v₂ / dt = m₂ d² r₂ / dt²

Here, I am being careful to distinguish between the forces on each mass as caused by the other mass and as caused by everything else. For example, the Earth and the Moon go around the Sun. The Moon is pulled on by the Earth and the Sun. The Earth is pulled on by the Moon and the Sun.

Now add the above equations. Remember Newton's third law? It says that if two objects act on each other then the forces are related by

F_{on 2, by 1} = - F_{on 1, by 2}

Now I'll manipulate the above force equations. ADD them.

F_{on 1, by 2} + F_{on 1, external} + F_{on 2, by 1} + F_{on 2, external} = m₁ d² r₁ / dt² + m₂ d² r₂ / dt²

The first and third terms cancel. Now manipulate the right-hand side. The masses are constants, so they can go either inside or outside the differentiation. Bring them inside and then multiply by one.

F_{on 1, external} + F_{on 2, external} = d² ( m₁ r₁ + m₂ r₂ ) / dt²

= ( m₁ + m₂ )

d²

dt²

m₁ r₁ + m₂ r₂

m₁ + m₂

This last expression defines the center of mass. This says that the final equation is really quite simple:

F_{total, external} = M_total a_CM

The motion of the center of mass behaves as a point with just the external forces being applied. None of the complications of the internal forces matter for this purpose.

Demo: Try this by throwing an off-center dumbbell. The center of mass is marked. It should follow a parabola.

11 Oct Do one of the homework problems in detail.

Show how the derivation of the center of mass acceleration in terms of the total external force can be modified and re-written in another form. This is

F_{total, external} = d p_total / dt,

where p_total is the "total momentum," and that is

p_total = p₁ +p₂ = m₁v₁ + m₂v₂.

The main reason that this form is useful is that when the total external force vanishes, then the total momentum stays constant. This is

Conservation of Momentum.

It applies a lot more often than you may think. Atomic and molecular collisions or collisions between galaxies are two extreme examples, but I showed the convervation law at work in a demo of two carts on a low-friction track.

13 Oct Do two of the homework problems in detail.

Derive the relation between force, displacemant and kinetic energy for the special case of constant force in one dimension. It's mostly a lot of routine algebra, eliminating the parameter, t, between two equations for position and velocity.

ma_x ( x - x₀ ) = F_x ( x - x₀ ) = mv_x²/2 - mv₀²/2

Watch out: that F_x is times the following factor, this is not a function of ( x - x₀ ).

I then stated the more general result that can be used for non-constant forces. Instead of deriving it, I had time only to show that it reduced to the correct special case if the force happened to be a constant.

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