Question

Show that Newton’s second law is the real law of motion.

Solution

Second law is the real law of motion:
This can be proved by showing that first law and third law are contained in second law of motion.
Let's prove that first law is contained in the second alw of motion.
According to the second law of motion,

   $straight F with rightwards arrow on top space equals space straight m straight a with rightwards arrow on top$
$space space therefore$ $straight F with rightwards arrow on top space equals space 0 space space space space space space rightwards double arrow space space space space straight a with rightwards arrow on top space equals space 0$
That is, if no force is acting on the body then its acceleration is zero. This means if a body is at rest, it remains at rest and if the body is moving in straight line with constant velocity, it continues to do so. This is what the first law states. Hence first law is contained in the second law.
(ii) Let's prove that Third law is contained in second law.
According to the second law of motion,
$space space space space space straight F with rightwards harpoon with barb upwards on top space equals space fraction numerator d p with rightwards harpoon with barb upwards on top over denominator d t end fraction i. e. comma space F with rightwards harpoon with barb upwards on top space equals space 0 space rightwards double arrow space p with rightwards harpoon with barb upwards on top space equals space c o n s tan t$
So, in th absence of external force, total momentum of the system remains constant.
Let, us consider an isolated system of two bodies of masses m₁ and m₂. Let, $straight F with rightwards harpoon with barb upwards on top subscript 12$ be the force exerted by mass m₁ on m₂ which is the reaction.
Let, $straight F with rightwards harpoon with barb upwards on top subscript 21$ be the force exerted by mass m₁ on m₂, which is the reaction.
Let, due to    $straight F with rightwards harpoon with barb upwards on top subscript 21$ ,
Rate of change of momentum of m₁ = $dp subscript 1 over dt$
Rate of change of momentum of m₂ = $dp subscript 2 over dt$
Therefore,
$space space space space space space space space space space straight F with rightwards harpoon with barb upwards on top subscript 12 space equals space fraction numerator d p subscript 1 over denominator d t end fraction space a n d space F with rightwards harpoon with barb upwards on top subscript 21 space equals space fraction numerator d p subscript 2 over denominator d t end fraction N o w comma space straight F with rightwards harpoon with barb upwards on top subscript 12 space plus space space F with rightwards harpoon with barb upwards on top subscript 21 space equals space fraction numerator d p subscript 1 over denominator d t end fraction plus fraction numerator d p subscript 2 over denominator d t end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator d over denominator d t end fraction open parentheses p subscript 1 space plus space p subscript 2 close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator d p with rightwards harpoon with barb upwards on top over denominator d t end fraction$
Here, $straight p with rightwards harpoon with barb upwards on top$ is the total momentum of the system.
According to the second law of motion, in the absence of external force, momentum remains conserved.
Therefore,
$therefore space straight F with rightwards harpoon with barb upwards on top subscript 12 space plus space straight F with rightwards harpoon with barb upwards on top subscript 21 space equals space fraction numerator straight d straight p with rightwards harpoon with barb upwards on top over denominator dt end fraction space equals space 0 rightwards double arrow space space straight F with rightwards harpoon with barb upwards on top subscript 12 space equals space minus straight F with rightwards harpoon with barb upwards on top subscript 21$
Hence, the third law of motion is proved.
Therefore, Newtons' second law of motion is the real law of motion.

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Laws Of Motion

Show that Newton’s second law is the real law of motion.

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