Newton's Laws of Motion

Kinetics is the study of how the motion of a body is related to its mass and the force(s) acting on it. The force represents the interaction of a body with its environment. The mass of a body is a measure of its inertia, which is the tendency to resist acceleration under a force. Newton's laws of motion are fundamental to the study of kinetics.

Newton's three laws of motion were first published in his celebrated work

Mathematical Principles of Natural Philosophy in the year 1687. We shall touch upon each of them.

Newton's first law of motion: A body at rest continues to remain at rest and a body in motion continues in motion with a constant velocity unless it is compelled by a net external force to change that state.

The statement of the first law provides two important concepts:

1. Inertia of a body and
2. Qualitative definition of force.

A body possesses two types of inertia, i.e. inertia of rest and inertia of motion. Inertia of rest is quite obvious since inanimate objects like the plates and the spoons on a dining table never begin to move on their own. However, inertia of motion is not so obvious in our daily lives. When a block is set in motion on the floor by giving it a push, it gradually loses speed and comes to rest on its own. But that is because of a resistive frictional force exerted by the floor on the block. If, ideally, the block would be set in motion on a perfectly smooth floor without friction, it would continue to move with the same velocity forever. Mass of a body is a measure of its inertia.

Newton's first law is often called the law of inertia. Any frame of reference in which Newton's first law is valid is called an inertial frame of reference. Qualitatively, a force is defined as an external agent which, when applied to a body, changes its state of rest or of motion with a constant velocity.

Newton's second law of motion: The rate of change in momentum of a body is proportional to the net external force that acts on the body and it takes place in the direction in which the force acts. The statement of the second law provides two more concepts:

1. Momentum of a body and
2. Quantitative definition of force.

Momentum is given by the product of the mass and velocity of a particle. Since mass is a scalar and velocity is a vector, momentum is a vector quantity which has the same direction as velocity. The dimensions of momentum of are [MLT-1], and its SI unit is kilogram metre per second (kg-m/s).

Quantitatively, force is given by the product of mass and acceleration. Since mass is a scalar and acceleration is a vector, force is a vector quantity which has the same direction as acceleration. The mathematical statement of Newton's second law is:

F = ma

This very important equation is called equation of motion. The dimensions of force are [MLT-2], and its SI unit is newton (N).

Newton's third law of motion: To every action there is always an equal and opposite reaction. In all cases, the action and reaction forces act on different bodies.

Whenever a particle 1 exerts a force of action, F₂₁, on another particle 2, the latter also exerts a force of reaction, F₁₂, on the former. These two forces are equal in magnitude and opposite in direction. They act along the same line joining the two particles. Therefore, we can write:

F12 = -F21

The above relationship is mutually reciprocal. That is, either of the two forces can be thought to be the action force and the other, the reaction force. Another important aspect of the third law is, even though forces always occur in pairs, they always act on two different bodies and do not cancel each other. If they were to act on the same body, the net force on the body would be zero and it could never have an acceleration!

Newton's second and third laws can be applied to explain a variety of events around us. But the first and foremost step towards that is to learn how to draw free body diagrams. Every time you want to write the equations of motion of a body, it is advised that you isolate the body from its surroundings, identify all the forces acting on it and show them clearly in a separate diagram. No need to show the bodies which exert the forces. The forces of reaction, which are exerted by the body under investigation, do not appear in the diagram. A force diagram of this type is known as a free body diagram. A correct free body diagram usually means correct equations of motion and a straightforward solution to the problem.

Among the many applications of Newton's second and third laws, problems on pulleys and wedges are of particular interest. The type of motion in which the velocities or accelerations of two or more particles depend on each other is known as constrained motion. When we analyse a constrained motion by applying Newton's laws, we must take into account additional conditions imposed on such a motion. This conditions are known as constraints. We shall discuss constrained motion of two types of system: pulley-block system and wedge-block system.

Any frame of reference, which accelerates with respect to an inertial frame of reference, is called a non-inertial frame of reference. Newton's laws are not valid in such a frame of reference. The net force acting on a particle in a non-inertial frame is made up of two parts: a real force and a fictitious force. The fictitious force, which has no material source, is called pseudo force and given by - ma₀ , where m is the mass of the particle and a₀ is the acceleration vector of the non-inertial frame with respect to the inertial frame. The pseudo force(s) do not exist when the motion is observed from an inertial frame.