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Let me make my question clear by the following example:- Let’s take a case of an individual who is in an elevator accelerating downward with some acceleration ‘ a ‘ such that ‘a’ < g . Let’s take the downwards direction as positive and let the normal force be ‘N’ the equation would then be mg - N = ma => N = m(g - a) Let’s say m = 60 kg , g = 10 m/s^2 , a = 5 m/s^2 Then, N = 60(10 - 5) => N = 300N , which does kind of make sense because you lose weight when accelerating downwards. But the question is why did my answer come out to be POSITIVE 300N , it should have been negative 300N , because I took the downward direction as positive. Isn’t this a mistake of me just assuming that ‘N’ = -N in my equation in the first place. This is wrong right, I cannot assume my force to already be negative,if it is truly negative, the answer will come out to be negative. Shouldn’t the equation then be N + mg = ma [ taking the same sign convention just taking N as N and not - N since Net force is = SUM of all the forces and not the difference] Then I would get N = m(a - g) = 60(5 - 10) which = - 300 N , which should be the correct answer. But every video I’ve seen on elevator physics just uses mg - N and gets a positive value of N but that’s wrong, isn’t it? The problem in this comes when we’re dealing , let’s say 2 masses fixed in a pulley such that m2 > m1enter image description here

Then solving :- I get: enter image description here I know that 5 m/s^2 is correct, but why did I get a positive answer because even the other mass is accelerating in the opposite direction. Yay did I get my acceleration which is correct for 1 mass? After solving, should I just say that that acceleration of the other = -5 just since it’s in the opposite direction without showing any calculation? Can you assume a negative sign on a force without knowing its value? I’m asking this because suppose you have a mass 5kg and acceleration: - 5 m/s^2 . Then to calculate force you would do f = ma = 5 x (-5)= - 50N and not : - F = ma => F = 5 x (-10)/-1 = 50N which is clearly wrong. So just because I know that my force is going to be negative,I can’t write -F = ma,instead,i just solve F =ma and automatically get my force to be negative which was my objective. Similarly,in kinematic equation problems,suppose u = 0 m/s a= -5 m/s^2 t = 5s v = ? I would do v = u + at = -5 x 5 = -25 m/s and not : -v = u + at Which would give me a wrong positive answer. So why can we assume a force to be negative?

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You're not "assuming the force is negative" here. You're choosing a convention for the direction of the force. This means associating the direction of the force with the sign of its magnitude. The convention you choose is entirely arbitrary and doesn't affect the physics. If you choose a different convention, then you get a number with the opposite sign at the end, but this still translates to the same direction.

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If you use -N, you are assuming the force is directed upward and are forcing a positive number for N. But it doesn't really matter what you assume: if you just use N, N comes out negative and if you use -N, N comes out positive. In either case, in the equation for the sum of forces on the person in the elevator the normal force contribution is -300 if you use the downward direction as positive (which you did by using +mg for the person's weight).

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