The previous page discussed momentum and how it is conserved in a closed system. This page is about how momentum can be exchanged between bodies by means of forces.
Not all forces transfer momentum, in some cases the forces may be in balance and therefore do not effect the dynamics. (see statics pages). If the forces are not in balance, then momentum will be transferred, this will create a force (inertia) which is equal and opposite to the sum of forces on the object.
In a collision between two objects the momentum can be exchanged very quickly (very large force in a very short time) in this case we call it an impulse as described here.
Newtons Second Law
A property of mass is its resistance to a change of velocity. This is quantified by Newtons second law which says that the acceleration is:
- Proportional to the net external force.
- Inversely proportional to its mass.
- In the direction of the force.
In other words:
F=m a
| where: | |||
symbol |
description |
type |
units |
| F | net force | vector | kg m/s2 |
| m | mass | scalar | kg |
| a | acceleration | vector | m/s2 |
There is also an equivalent for rotation:
| τa = [Ia]α or |
in inertial fixed frame (varies with time) |
| τr = [Ir]α + ω×[Ir]ω | in body coordinates (inertia tensor fixed) |
| where: | |||
symbol |
description |
type |
units |
| τa | torque in absolute coordinates (body being moved relative to ground) | bivector | N m |
| τr | torque in relative coordinates (control surfaces or jets on body) | bivector | N m |
| [Ia] | inertia tensor in absolute coordinates (this will vary with orientation and since the body is rotating the elements of the tensor will vary with time) | tensor | kg m2 |
| [Ir] | inertia tensor in relative coordinates (since this is relative to the body it will be constant) | tensor | kg m2 |
| α | angular acceleration | bivector | s-2 |
The first form of the equation looks simplest but the values vary with time and therefore the second form can be easier to use. Since it is, generally, very difficult to continuously recalculate the changing inertia matrix as seen from an inertial fixed frame, the conservation of angular momentum is most frequently written in a coordinate frame fixed in the rotating rigid body.
If the rotation vector is aligned with a principal axis of the rotating body then the term: ω×[Ir]ω becomes zero, the term: ω×[Ir]ω describes the behavior known as gyroscopic or processional motion.
Tensor
 -
A set of components that obeys some transformation law in n-dimentional
space. more |
When working in 3 dimensions, then F, a, T and 
can be represented as vectors and m and I as matrices.
For example, then the equations become:
| F=m a |
|
|||||||||||||||||||
| τ = [I] α |
|
where x,y,z are the mutually perpendicular coordinate directions. Note that mass, m is the same in all directions whereas the coordinate