Question

State Newton’s second law of motion. Derive a mathematical expression for Newton’s second law.

Answer 1

Newton's Second law of motion :- The rate of change of momentum is directly proportional to the force applied on the system.

Force applied is directly proportional to the product of mass and acceleration .

Let $p_{i} , p_{f}$ be the initial and final momentums respectively.

According to newton's second law :-

pf - pi / t ∝ F

We know that,
Momentum ( P) = mv .

Let v be the final and u be the initial velocity .

Now,

mv - mu / t ∝ F

F ∝ m ( v-u) /t

F ∝ ma.

F = kma.

Here, K is the proportionality constant. It's value is 1 .

Units of Force are given by the units of mass and acceleration. Units of force is Kgm/s² .

In accordance to honour the contributions of Newton, 1 kgm/s² is termed as 1 Newton.

Answer 2

Hi,

Here is your answer,

This law states that ''the rate of change of momentum of a body is directly proportional to the external force applied on the body and the change takes place in the direction of the applied force''.

Let F be the external force applied on the body in the direction of motion of the body for time internal Δt, hen the velocity of a body of mass 'm' change form v to v Δv change in momentum Δp = mΔv.

Now, According to Newton's Law of Motion

                    F α p/ t  or F = k  p/ t 

where k is a constant of proportionality 

If limit Δt = 0, then the term Δp/ t becomes the derivative dp/dt

Thus,                         F = k dp/dt

For a body of fixed mass m, 

                F = k d(mv)/dt = km dv/dt

               F = kma                              (∴ dv/dt = a)

Let,             K = 1

                    Force, F = ma

In scalar form, it can be written as

                     F = ma

∴ 1 unit force = 1 unit mass × 1 unit acceleration 

↔ A unit force may be defined as the force which produces unit acceleration in a body of unit mass.

The force is a vector quantity and its SI unit is Newton.



Hope it helps you !