We know that,
F ∝ v²/r.
Also, from Newton's second law of motion, the force applied on an object is proportional to the change in momentum.
=> F ∝ ∆p/t
=> F ∝ (p2 – p1)/t
=> F ∝ (mv – mu)/t
=> F ∝ m(v–u)/t
=> F ∝ ma.
So, F ∝ v²/r and F ∝ ma means that v²/r ∝ ma.
=> v² ∝ mar.... [equation (1)]
Now, for an object moving in a circular path, velocity = distance/time => v = 2πr/t
=> v² ∝ r²/t² (since 2π is a constant)
=> mar ∝ r²/t²
=> mart² ∝ r²
=> Frt² ∝ r²
=> π × Frt² ∝ π × r²
=> πFrt² ∝ area of the circular path around which the object is moving (since area of any circular path = π × r² = πr²).
Transposing π from LHS to RHS, we get
=> Frt² = 1/π × area of the circular path (since 1/π = constant).
=> Frt² = (area of the circular path around which the object is moving)/π.... equation (2).
But, if this so, then the units of LHS and RHS should match at last, i.e., unit of Frt² = unit of (area of circular path)/π.
But, we know that this is not the case, because unit of Frt² = kg m/s² × m × s² = kg m², and unit of (area of circular path)/π = m².
Then, is equation (2) mathematically wrong?
Answers
Answer:
A very good analytical approach.. but there is a problem in your first assumption.. the problem is of the following nature , the two forces you are considering are of different nature.. the equation F = mv^2 / r , is centripetal force which is applicable only for curved trajectories and this force has its own special properties like it cannot change the magnitude of the velocity and so on.. whereas f = ma is more general in nature and more fundamental , so at the very beginning you cannot equate these two forces.. secondly you have made mistakes in putting proportionality constants.. when u remove the proportional sign and place a equality u need to put a constant.. for a moment if we assume whatever u have assumed in the first scenario is true then the proportionality constants will take care of the units.. Hope this helps..