# Newtonian Mechanics

### I. Definitions.

Firstly let us define the following quantities:

- 1.
**Force.**- a push or a pull. - 2.
**Speed.**- distance covered per unit of time. - 3.
**Velocity.**- speed in a given direction. - 4.
**Uniform Velocity.**- constant speed in a straight line. - 5.
**Acceleration.**- change of velocity per unit of time.

### II. Gravitation.

Newton's work on Gravitation can be summarised as follows:

*(i) every body in the universe attracts every other body *

*(ii) the gravitational force between two bodies is directly proportional to the mass of each and inversely proportional to the square of the distance between them.*

### III. Motion.

Newton's work in mechanics can be summarised in a statement of his three laws. The first tells us what happens in the case of a body on which the net force is zero.

*I. Every body continues in a state of rest or of uniform velocity unless acted on by an external force.*

The second law tells us how to deal with bodies that have a non-zero net force:

*II. The acceleration of a body under the action of a net force is directly proportional to that force and inversely proportional to the mass of the body. *

In mathematical terms, if **F **is the force, **m **the mass and **a **the acceleration, Newton' Second Law can be stated succintly as **F = ma **- probably the most famous equation in all of Physics!

The third law talks about the mutual forces that two bodies in contact exert on each other, and can be stated thus:

*III. To every action there is an equal and opposite reaction.*

The third law will not have much impact on the progress of this course, so we will not consider it further. However, if this confuses you (*e.g. how can a body ever move if it experiences equal and opposite forces?*) remember that the action and the reaction mentioned in the third law act on **different** bodies - I push on the earth (*action*) and the earth pushes back on me (*reaction*).

To give you an idea of the application of this new understanding of mechanics, let us look at the old question of whether, when two bodies are dropped from the same height, the heavier would reach the ground first, a question answered in the affirmative by Aristotle.

Let us consider the case of an elephant falling out of a tree. As Galilieo showed, every falling body experiences the same acceleration in the absence of air resistance. However, when there is air resistance, the situation changes. Initially the elephant experiences only the force due to gravity, pulling it towards the centre of the earth. As its speed increases, however, air resistance also increases, opposing the force of gravity, which does not change. Eventually the force of the air resistance upwards equals the force of gravity downwards so that the net force on the elephant becomes zero. Newton's first Law then tells us that the elephant will continue from that point on with a constant speed. This speed is called the terminal velocity.

Now suppose that a feather drops from the tree at the same time as the elephant. In this case the gravitational force on the feather will be much less than it was on the elephant. So the air resistance on the feather will very quickly become equal to the force of gravity and the feather's terminal velocity will be much smaller than that of the elephant. Thus, in this case, Aristotle is correct, and the elephant will reach the ground long before the feather. Of course, in the absence of air resistance the situation is quite different. Although the force of gravity on the elephant is much greater than that on the feather, the elephant has a much greater inertial mass; and since acceleration is inversely proportional to the mass, it turns out that the acceleration of both is identical.