- #1

ksy

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- TL;DR Summary
- In Newton's second law problems, some textbooks (like Halliday and Resnick) substitute "a" for a component of the acceleration, irrespective of its sign. Why is that possible?

Hi,

I was looking over one of the sample examples in Halliday and Resnick, the one

about the scale in the elevator. There is something that bugs me about it, and I'd

like to know if you agree.

The example has to do with finding the reading of a scale that is measuring

someone's weight in a moving, and especially accelerating elevator (where

the acceleration is measured in the ground frame to use Newton's 2nd law).

This reading is the same as the absolute value of the normal force from the person on the scale, and

therefore the absolute value of the normal force from the scale on the person,

from Newton’s third law.

The positive y direction is chosen to be upwards, as usual.

Newton's second law is then taken to be:

Surely that’s the equation for the y components of the vectors involved:

But at this stage, we don’t know the direction of the acceleration, so why are we

implicitly stating that
?

Then, substituting mg for
that leads to:

And so far, nothing has been said about whether the elevator is accelerating up or

down. So we don’t know whether the y component of a is positive or negative.

Now the example goes on to say that for the elevator accelerating up at 3.20 m/s^2 ,

a = + 3.20 m/s^2 so:

N = (72.2kg) (9.8 m/s^2 + 3.20 m/s^2 )

While if the elevator accelerates down at 3.20 m/s^2 ,

a = - 3.20 m/s^2 so:

N = (72.2kg) (9.8 m/s^2 - 3.20 m/s^2 )

To me, this only makes sense if
that is a

components can be positive or negative, while magnitudes are always positive.

And obviously at this point we are dealing with an algebraic equation, not a vector

one so we can’t mean

I’ve seen this done since I was in high school, when we were told “if you assume

the wrong direction for a, it will just come out negative”. I’ve seen it taught that

way in colleges and universities. And I’ve done it hundreds of times out of habit.

But now that I think about it, I can’t help wonder: how can assuming that

how can that not violate some kind of homogeneity and still be correct? What is "a"

in there?

If we want to get rid of the y subscript on "a" without first knowing in which

direction

No??? Or what am I missing?

Thank you for your thoughts!!!

I was looking over one of the sample examples in Halliday and Resnick, the one

about the scale in the elevator. There is something that bugs me about it, and I'd

like to know if you agree.

The example has to do with finding the reading of a scale that is measuring

someone's weight in a moving, and especially accelerating elevator (where

the acceleration is measured in the ground frame to use Newton's 2nd law).

This reading is the same as the absolute value of the normal force from the person on the scale, and

therefore the absolute value of the normal force from the scale on the person,

from Newton’s third law.

The positive y direction is chosen to be upwards, as usual.

Newton's second law is then taken to be:

Surely that’s the equation for the y components of the vectors involved:

But at this stage, we don’t know the direction of the acceleration, so why are we

implicitly stating that

Then, substituting mg for

And so far, nothing has been said about whether the elevator is accelerating up or

down. So we don’t know whether the y component of a is positive or negative.

Now the example goes on to say that for the elevator accelerating up at 3.20 m/s^2 ,

a = + 3.20 m/s^2 so:

N = (72.2kg) (9.8 m/s^2 + 3.20 m/s^2 )

While if the elevator accelerates down at 3.20 m/s^2 ,

a = - 3.20 m/s^2 so:

N = (72.2kg) (9.8 m/s^2 - 3.20 m/s^2 )

To me, this only makes sense if

*component*of a. Becausecomponents can be positive or negative, while magnitudes are always positive.

And obviously at this point we are dealing with an algebraic equation, not a vector

one so we can’t mean

**a**as a vector.I’ve seen this done since I was in high school, when we were told “if you assume

the wrong direction for a, it will just come out negative”. I’ve seen it taught that

way in colleges and universities. And I’ve done it hundreds of times out of habit.

But now that I think about it, I can’t help wonder: how can assuming that

how can that not violate some kind of homogeneity and still be correct? What is "a"

in there?

If we want to get rid of the y subscript on "a" without first knowing in which

direction

**a**points, surely we should write:No??? Or what am I missing?

Thank you for your thoughts!!!