Question

State Newton’s second law of motion. Derive a mathematical formula to calculate force if mass remains constant. Prove that it contains Newton’s first law.

Answer 1

We know that the force applied on any object makes certain physical and visible changes. There are many daily life examples which relate the laws of motion. Here, the second and first law will help us to find the reason for the above question.

Formula used:
\eqalign{& F = \dfrac{{dp}}{{dt}} \cr & p = mv \cr}

Complete step-by-step answer:
Newton's Second law of motion is given as the rate of change of momentum is directly proportional to the force applied on the system.
It also gives us a relation between force F, mass m and acceleration of an object. Force applied is directly proportional to the product of mass and acceleration.
Riding a bicycle is a good example of this second law of motion at work. If we consider that our bicycle is the mass. Our leg muscles push the pedals of our bicycle. When we push on the pedals, our bicycle accelerates. Therefore this shows that we are increasing the speed of the bicycle by applying force to the pedals.
When the mass m is constant, the force applied on an object is given by:
\eqalign{& F = \dfrac{{dp}}{{dt}} \cr & \because p = mv \cr & m = const. \cr & \dfrac{{dp}}{{dt}} = m\dfrac{{dv}}{{dt}} + v\dfrac{{dm}}{{dt}} \cr & \Rightarrow \dfrac{{dp}}{{dt}} = m\dfrac{{dv}}{{dt}} \cr & \therefore F = m\dfrac{{dv}}{{dt}} \cr}
So, we get the expression of force, when the mass of an object is constant.
Now, as we know Newton’s first law of motion, states that an object continues to be in rest or in motion in a particular direction until and unless any external force is applied on it, this is also known as law of inertia.
Inertia is the tendency of an object to remain at rest or in motion; it is related to mass of an object.
Therefore, in the above expression of force, where mass is constant, also proves the first law of motion, which is also called law of inertia.

Additional Information: Force is simply any push or pull. The S.I unit of force is Newton represented by N. The acceleration is defined as the increase in the velocity of an object. The acceleration is measured in meters per Second Square.
There are three laws of motions given by Newton. These laws of motion relate an object's motion to the forces acting on it.
First law of motion states that an object continues to be in rest or in motion in a particular direction until and unless any external force is applied on it.
In the second law of motion, the force on an object is equal to its mass times its acceleration. This law also gives the relation of momentum and force.
In the third law of motion, every action has an equal and opposite reaction.

Note: Force applied on an object changes its motion, speed, direction and also its shape. There is a very small difference between the force applied and the pressure applied on an object. Pressure is always given as the force per unit area.