Derive the mathematical form of second law of motion
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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 . ... We know that, Momentum ( P) = mv . Let v be the final and u be the initial velocity .
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the second law of motion states that the rate of change of momentum is directly proportional to the applied unbalanced force
The initial and final momentum of the object will be, p1 = mu and p2 = mv respectively.
The change in momentum ∝ p2 – p1
∝ mv – mu
∝ m × (v – u).
The rate of change of momentum ∝ m × (v −u) / t
Or, the applied force,
F ∝m × (v −u) / t
Or, the applied force, F = km × (ν −u) / t ----------- (2
= kma ------------(3)
Here [a = (v – u) / t ] is the acceleration, which is the rate of change of velocity. The quantity, k is a constant of proportionality. The SI units of mass and acceleration are kg and m s-2 respectively. The unit of force is so chosen that the value of the constant, k becomes one. For this, one unit of force is defined as the amount that produces an acceleration of 1 ms-2 in an object of 1 kg mass. That is,
1 unit of force = k × (1 kg) × (1 m s-2)
Thus, the value of k becomes 1. From Eq. (3) , therefore, F = ma ---------(4)
- The unit of force is kg m s-2 or newton, which has the symbol N. The second law of motion gives us a method to measure the force acting on an object as a product of its mass and acceleration.
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- The first law of motion can be mathematically stated from the mathematical expression for the second law of motion. Eq
F = ma
or F= m (v- u) / t --------------(5)
or Ft = mv – mu
That is, when F = 0, v = u for whatever time, t is taken.
- This means that the object will continue moving with uniform velocity, u throughout the time, t. If u is zero then v will also be zero. That is, the object will remain at rest.