Back to: Further Mathematics SS2

**Sir Isaac Newton** put forward three important laws which relate to the motion of bodies under the action of given forces. These laws are central to the study of dynamics, since **dynamics **essentially involves the study of motion of bodies under given forces

**The Frist Law OF Motion**

If we place a ball on the ground, it will continue to rest there until someone comes to kick it. Once it is kicked, it will start moving and continue to move until something happens either to stop it or change its direction of motion. This basic idea is stated in Newton’s First Law which may be stated as: **Everybody continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by external impressed forces.**

This law re-emphasizes the fact that a force can change the state of rest or uniform motion of a body. A stationary point will remain stationary unless it is pushed from its stationary position. By pushing, we are exerting a force on the object. A moving car will continue to move unless brakes are applied to bring it to a halt. The brakes applied have introduced a kind of force that makes the car to come to a stop.

The tendency of a body to remain in its state of rest or uniform motion in a straight line is called **inertia **and is a function of the mass of a body. The greater the mass of a body, the greater its inertia and hence the greater the force required to change the state of the body.

**The Second Law of Motion**

The law states that **the rate of change of momentum of a body is proportional to the applied force and is in the direction of the force.**

The second law of motion helps us to obtain an expression for 5the force acting on a body. We recall that momentum is defined as the product of mass and velocity.

By the second law of Newton

The law established an exact relationship between force *F, *the mass *m *of a body, as well as the acceleration **a** of the body.

If a force* F*acts on a body of mass m kg it produces an acceleration in the mass given by the relation

**F = ma**

Newton’s second law also enables us to deduce the unit of force. We recall that the unit of mass is kilogramme (kg). The unit of mass is meter per second ( ). Hence, the unit of force is kg .

A force acting on a body of mass 1 kg, producing an acceleration of 1 is called **1 Newton (1N).** So the unit of force is the Newton.

**Newton’s Third Law of Motion**

Newton’s third law of motion states: ** Action and reaction are equal and opposite. **When two bodies are in contact, the forces of action and reaction are equal in magnitude and opposite in direction.

Such forces are also collinear. Let us consider a heavy block placed on a table, the force due to gravity on the body (weight of the block) acts directly on the table downwards. The table will have to exert an equal but opposite force on the block. This force acts upwards and balances the weight of the block on the table. If the table cannot withstand the weight of the block, it collapses.

**Example 1**

A boy sits on a log. The mass of the log is 8 kg and the weight of the boy is 55N. What is the reaction of the ground on the log on which the boy is sitting? (Take g= 9.8 )

**Solution**

Weight of the log = 8 9.8N

=78.4N

Weight of the boy and the log = (78.4 + 55) N

= 133.4N

By the third law of Newton, the ground will expert an equal but opposite force on the log on which the boy is sitting.

Hence if *R* is the force reaction of the

Consider a body of mass m on a smooth plane inclined at angle θ to the horizontal.

The force on the body due to gravity (weight) acts vertically downward and is *mg.*The force which acts perpendicularly to the inclined plane in *mg *cosθ .

The reaction of the inclined surface on the body is *R *and is equal in magnitude to *mg. * The force which tends to move the body down the plane is *mg*sinθ. The force which tends to move the body up the plane is *F* –mg sinθ. The equation of motion is:

*F* – mgsinθ = ma

Where a is the acceleration of the body. If however, F <mgsinθ then the body will move down the plane with a net force of *mg*sinθ – F = ma where a is the acceleration of the body down the plane.

**Example 5**

An object whose weight is 10kg is placed on a smooth plane inclined at 30 to the horizontal. Find:

a) the acceleration of the object as it moves down the plane;

b) the velocity attained after 3 seconds if:

(i) it starts from rest;

(ii) it moves with an initial velocity of 5

[Take g = 10

**Example 6**

A body of mass m is placed on the surface of a smooth plane which is inclined at an angle θ to the horizontal. A force f whose line of action is parallel to the surface of the inclined plane acts on the body to just prevent it from slipping down the plane. If R is the reaction between the surface of the inclined plane and the body, show that F = R tanθ

**Solution**

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