Properties of sentre of mass

Let us rewrite equation for centre of mass in the following form:

The left part of this equation includes product of the total mass of all particles constituting the system and velocity of centre of mass. The right part includes algebraic sum of all particles. Then the total momentum of the system of all particles can be written as:

It means that there is such dependence:

the momentum of the system of particles equals
the momentum of centre of mass

Thus, to consider the motion of only one material point instead of system of particles is enough. Mass of the point equals the sum of the masses and velocity equals velocity of centre of mass.

Motion of material point is described by Newton’s laws of dynamics.

If external forces are absent (the system is closed), total momentum must be constant. Therefore velocity of centre of mass remains constant.

Newton’s first law is valid for centre of mass

Now we’ll find the rate of change of total momentum:

.

where is acceleration of centre of mass.

Rate of change of momentum of material point equals external force , therefore

Acceleration of centre of mass is proportional to external force and it has the same direction.

Newton’s second law is valid for centre of mass

It is necessary to distinguish sentre of mass and sentre of gravity, the coordinate of which is determined according to formula:

where G_i is the force of gravity of particle with coordinate .

If the gravitation field is homogeneous, free fall acceleration for all particles is identical. At this case center of gravity coincides with the center of mass:

In the heterogeneous gravitation field free fall acceleration of each particle requires different values and center of gravity does not coincide with the center of mass.

3.1.2. Systems with varying mass: a rocket

Fig. 3.2

If we divide a body into two parts and one of them will move at velocity , the second part of this body will move at velocity according to momentum conservation law (Fig. 3.2).

We’ll take the center of mass of a composite body as reference frame.

External forces are not present. The total momentum of compositebody equals zero:

After the body is divided the total momentum remains constant and equals zero:

Knowing the velocity of the first body, we can find the velocity of the second one:

As no forces act on bodies, flying away, their velocities will remain constant.

We have assumed that the total mass of the system remains constant in the systems we have dealt with so far. Sometimes it does not remain constant as in the case of a rocket. Most of the mass of a rocket standing on its launching pad is fuel, which will burn and the gases are ejected from the nozzle of the rocket engine.

Let velocity of a rocket with mass m at the moment of time t equals (Fig. 3.3). is the velocity of exhaust product relative to the rocket. is the velocity of exhaust product relative to the Earth. Momentum of a rocket at the moment of time t is

Fig. 3.3

As velocity and mass of a rocket is changed, the rate of change of momentum of rocket is:

All velocities have the same direction, therefore vectors are omitted.

The rate of change of the exhaust product is

where dm is the mass of the exhaust product.

According to Newton’s third law the rate of change of a rocket momentum equals the rate of change of momentum of the exhaust product taken with sign “minus”:

або

We’ll transform this expression as follows:

,

or

So the differential equation of motion of a body with changeable mass was obtained. The left part includes product of mass and acceleration which is force. It is reactive force. It is determined by velocity of the exhaust product:

Consequently, reactive force will act on body with changeable mass and it will move with acceleration.

We’ll rewrite the differential equation of motion in the following form:

It is the equation of Meshchercky.

Having integrated equation of Meshchersky we’ll obtained:

, ,

or

The equation above is called еquation of Tsiolkovsky.

Here we see the advantage of multistage rockets. Final rocket mass decreases with discarding successive stages when only fuel is depleted. An ideal rocket would reach its destination with only its remaining pay load.

3.2. CONSERVATION OF ANGULAR MOMENTUM

We start with Newton's second law in angular form.

If no net external torque acts on the system, this equation looks like:

or

This result, called THE ANGULAR MOMENTUM CONSERVATION LAW, can also be written as

where the subscripts refer to the values at some initial time t_i and later time t_f. Thus

If the net external torque acting on a system is zero,
the angular momentum of the system remains constant,
no matter what changes take place within the system

Equations of the law of conservation of angular momentum are vector equations; so, they are equivalent to three scalar equations corresponding to the conservation law of angular momentum in three mutually perpendicular directions. Depending on the torques acting on a system, the angular momentum of the system might be conserved in one or two directions but not in all directions:

If any component of the net external torque of a system is zero,
then that component of the angular momentum of the system along that axis
cannot be changed, no matter what changes take place within the system

We can apply this law to the isolated body, which rotates around the z axis. Let’s suppose that the initially rigid body redistributes somehow its mass relative to this rotation axis, changing its rotational inertia. Equations of the law of conservation of angular momentum state that the angular momentum of the body cannot be changed. Taking into account that (for the angular momentum along the rotational axis), we write this conservation law as

Here the subscripts refer to the values of the rotational inertia I and angular speed before and after the redistribution of mass.

Angular momentum conservation law holds beyond the limits of Newtonian mechanics. It holds for particles whose speeds approach that of light (where the theory of relativity is valid), and it remain true in the world of subatomic particles (where quantum mechanics is valid). No exceptions to the law of conservation of angular momentum have ever been found.

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