# Standing on a scale on an elevator is a man of mass 140kg. a) what is the reading of the scale if the elevator moves up at 5 m/s. b) moves down at 5m/s c) accelerated up at 2m/s^2 d) accelerates down at 2m/s^2?

for a) and b) would I use N=m(g+a) but when I did I got the same answer would that be right since a=a in a) and b). For c) and d) I would use the same equation as a) and b) but the only difference would be the acceleration for c) which would be positive 2 and d) would be negative 2.

You mean a = g, in problems a) and b),? Yes that's right. And the answer for c) and d) also correct.

It's still instructive to actually derive the formula N=m(g+a)...

## To derive the formula N = m(g + a) for the reading on the scale, let's break it down step by step:

1. Start with Newton's second law: F = ma, where F is the net force applied to an object, m is its mass, and a is the acceleration.

2. Consider the forces acting on the man standing on the scale in the elevator. The three relevant forces are:
- The man's weight, W = mg, which always acts downward.
- The normal force, N, exerted by the scale on the man, which acts upward.
- The force due to the acceleration of the elevator, which we can call F_acc.

3. In case a), where the elevator moves up at a constant velocity, there is no acceleration. Thus, F_acc = 0, and we only have two forces acting: W and N. The man experiences an apparent increase in weight as he is pushed against the scale due to inertia. So we have:
- N - W = 0.

4. Rearrange the equation to solve for N. Since W = mg, we get:
- N = mg.

5. In case b), when the elevator moves down at a constant velocity, we have the same forces acting as in case a), but the man experiences an apparent decrease in weight due to inertia. So we have:
- N - W = 0.

6. Rearrange the equation to solve for N, similarly to step 4, which gives us:
- N = mg.

Therefore, in both cases a) and b), the reading on the scale is equal to the man's weight, N = mg.

Now let's consider cases c) and d), where the elevator is accelerating:

7. In case c), when the elevator accelerates upward at 2 m/s^2, we have three forces acting: W, N, and F_acc. The equation based on Newton's second law becomes:
- N - W = ma.

8. Rearrange the equation to solve for N, and substitute W with mg, gives us:
- N = mg + ma.

9. Simplify the equation to factor out m, which yields:
- N = m(g + a).

So, in case c), the reading on the scale is given by N = m(g + a), where a is the upward acceleration of the elevator.

10. In case d), when the elevator accelerates downward at 2 m/s^2, the forces acting are the same as in case c), but now the acceleration is negative. The equation becomes:
- N - W = ma.

11. Rearranging the equation and substituting W with mg results in:
- N = mg - ma.

12. Simplify the equation to factor out m, which gives:
- N = m(g - a).

Thus, in case d), the reading on the scale is N = m(g - a), where a is the downward acceleration of the elevator.

In summary, for cases a) and b), the reading on the scale is equal to the man's weight, N = mg. For cases c) and d), the reading on the scale is given by N = m(g + a) and N = m(g - a), respectively, where a is the upward or downward acceleration of the elevator.