DYNAMICS :Newton’s Laws Of Motion, Motion Along Inclined Plane And Motion Of Connected Particles

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 Facts 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 (𝑚𝑠⁻²). Hence, the unit of force is kg

𝑚𝑠⁻².

A force acting on a body of mass 1 kg, producing an acceleration of 1 𝑚𝑠⁻² 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= 9.8𝑚𝑠⁻²)

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.

MOTION ALONG AN INCLINED PLANE

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 mgsinθ. 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 mgsinθ – F = ma where a is the acceleration of the body down the plane.

MOTION OF CONNECTED PARTICLES

In this unit, we shall examine the motion of two or more bodies connected by a light inextensible string, connected to a light smooth pulley. The basic assumption we make is that the tensile force in the string is always the same throughout every section of the string. We can easily write down the equation of motion of the connected particles, once they are set in motion.

EVALUATION

A force P acts on a body of mass 5kg on a smooth horizontal floor if it produces an acceleration of 4.5 m/s , find the magnitude of P

GENERAL EVALUATION

1. A body of mass 15kg is placed on a smooth plane which is inclined at 60 to the horizontal, find the acceleration of the body as it moves down the plane
2. A body of mass 5kg is connected by a light inelastic string which is passed over a fixed frictionless pulley by a movable frictionless pulley of mass 1kg over which is wrapped another light inelastic string which connects masses 3kg and 2kg , find the acceleration of the masses and the tension in the strings

Reading Assignment

New Further Maths Project 2 page 237- 242

WEEKEND ASSIGNMENT

A body of mass of mass 100kg is placed in a lift , find the reaction between the floor of the lift and the body when the lift moves upward

1. at constant velocity a) 800N b) 900N c) 1000N d) 600N
2. with an acceleration of 3.5m/s a) 100N b) 1350N c) 1200N d) 1500N
3. A body of mass 20kg is placed in a lift , find the reaction between the floor of the lift and the body when the lift moves downward with a retardation of 2.5 m/s a) 250N b) 300N c) 350N d) 400N
4. Law of inertia is also known as Newton”s —– Law of motion    a) 2^nd b) 1^st c) 3^rd d) 4^th
5. The relationship between force and acceleration of a body in motion can be attributed to Newton”s  ——–  Law of motion    a) 1^st b) 2^nd c) 3^rd d) 4^th

THEORY

1. A car of mass 0.9 tonnes is moved by a constant force F from a speed of 12m/s to 16m/s over a distance of 50m, find F
2. Two masses 10kg and 8kg are connected by a light inextensible string which is passed over a light frictionless pulley fin the tension in the string