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Reading

Fowles and Cassidy 7.5 - 7.7.

Recall

We've seen that the motion of a system of particles can be broken up into 3 parts:

1. The point-like motion of the center of mass (CM) of the entire system governed by forces external to the system.
   F_tot = å _i F_i = dp_CM/dt
2. Rotation about the CM governed by external torques.
   dL_tot/dt = N_tot = å _i r_i ´ F_i
   L_tot = L_CM + L_rel
3. Internal motion governed by interactions between particles in the system.

The overall idea when analyzing the motion of a system is to take a top down approach. Start with the motion of the CM of the system. Then look for rotations about the center of mass (a collective motion). Finally, consider the individual motions of particles relative to the CM.

Kinetic Energy

For a system of n particles the kinetic energy is the sum of the individual kinetic energies:

T_tot = å _i T_i = å _i 1/2 m_iv_i²

Now if we express v_i in terms of the CM velocities, we can break the kinetic energy into a piece due to the motion of the CM and a piece due to the relative motions of the particles.

v_i² = (v_CM + v'_i)· (v_CM + v'_i) = v_CM² + v'_i² + 2v_CM· v'_i

Inserting this into the kinetic energy expression, and noting that the remaining dot product sums to zero, we get:

 T_tot = 1/2 Mv_CM² + å _i 1/2 m_iv'_i² = T_CM + T_rel

 In words, the total kinetic energy of a system is equal to the kinetic energy due to the motion of the CM plus the kinetic energy due to motion of particles of the system relative to their CM. This latter part can come from rotation of the system (a collective motion of all the particles of the system), or the motion of individual particles of the system.

Example: Kinetic Energy of the Swinging Rod

Let's calculate these terms for the case of the swinging rod that we looked at in the last lecture.

T_CM = 1/2 mv_CM² = 1/8 mw ²l²

T_rel = å_i 1/2 m_iv'_i² = w ²/2 å_i m_ir'_i² Þ w ²/2 ò r²dm = 2w ²/2 ò ₀^l/2 r²dm = (1/24) mw ²l²

T_tot = w ²/2 ò ₀^l r²dm = (1/6) mw ²l²

A quick check confirms that T_tot = T_CM + T_rel.

Collisions

Collisions are an important tool in physics. Much of our understanding of the subatomic world comes from the study of collisions between subatomic particles. Many aspects of the motion of comets and man-made satellites can be understood as "collisions" with planets and stars. And even the physics of semiconductors and superconductors is phrased in the language of collisions.

In a collision, two objects approach each other, interact, and then continue with their motion. We generally assume that their interaction occurs over a finite time and distance, then, even without knowing the details of their interaction (gravitational, electrical, or any other 'al) we can understand the kinematics of the collision. The reason is conservation of momentum and energy.

1. In the usual situation, there are no external forces on the two objects. Therefore, the total momentum of the two objects is conserved. This is always true independent of whether the forces between the objects are conservative or not.
2. Total energy is conserved in the collision, but sometimes the collision process transforms some of the energy from mechanical to other forms, for instance heat. We quantify the degree of energy conservation by the quantity e = D v'/D v. If e =1, mechanical energy is conserved, and we say that the collision is elastic. For e ¹ 1, mechanical energy is not conserved, and we say that the collision is inelastic.

Example: Double Ball Drop

Impulse

Variable Mass Motion: Rockets

Consider a rocket lifting off from earth. The only external force on the rocket is gravity, which tends to hold the rocket on the ground. How does a rocket lift off and move hundreds or thousands of kilometers from earth?

Traditional rockets move by ejecting a part of their mass at high speed opposite the desired direction of motion (really force, but motion sounds better).

Consider two objects initially moving separately, at different velocities, but then combining into one a short time later (this is the time reversed picture of a rocket, as done in the text). Initially the momentum is P_t = mv + uD m. After the two objects combine they are one object moving at a new velocity P_{t+D t} = (m+D m) (v+D v).

Ó 25 February, 1999 R. Harr