Understanding the Direction of Acceleration in SHM: Mathematically Explained

[koliko987]

Mathematically, in SHM,why is x'' (acceleration) always in the direction if x increasing? So if he have a simple setup, an elastic spring on a smooth horizontal table, one end attached to a fixed point, the other to a particle. Let's say the fixed point is at the left end of the spring. If we take right as positive and we pull the particle to the right, stretching the spring, the tension acts to the left and hence the acceleration must be acting in the same direction, from Newton's 2nd law, OPPOSITE to the direction of x increasing since that's to the right. But my book and other resources I found say "acceleration is always in the direction of x increasing." So what does that mean?
I mean if I just see the shm equation it does make sense that acceleration/force has a different sign to the displacement and acts to restore the particle to equilibrium and if we look at the graphs, acc graph is 180 degrees out of phase to the displacement graph so surely this means they always act in opposite direction? Why does my book keep insisting acc is always in the direction of positive x (that's what x increasing means, right?).
I'm rally getting frustrated and any help is hugely appreciated.

 

[Delta2]

Well you are right that acc and displacement are 180 degrees out of phase.Not sure what your book is trying to say but seems wrong to state that acceleration is always in the direction of x increasing. This is true during two quarters of the harmonic circle, but during the other twp quarters acceleration is in the opposite direction of x increasing.

 

[koliko987]

Well you are right that acc and displacement are 180 degrees out of phase.Not sure what your book is trying to say but seems wrong to state that acceleration is always in the direction of x increasing. This is true during two quarters of the harmonic circle, but during the other twp quarters acceleration is in the opposite direction of x increasing.

It was brought to my attention that physical acc might not be the same as mathematical acc and mathematical acc is really the second derivative of x with respect to time, and the book is saying x˝ (not acc) is in direction x increasing, so I don't know if that makes a difference. And as far as the "physical" acceleration is concerned I mean it definitely changes direction depending on which side of the equilibrium point the particle is so surely it would be wrong to say that acc is in the same anyone direction during whole of oscillations. However, the book is adamant about " x˝ always in the direction of x increasing ", it mentions it 5 or 6 times and the book is highly regarded in UK so I doubt it's wrong.

 

[sophiecentaur]

Acc is the same Acc, wherever you come across it. (How could it be otherwise if the Maths is used to model the reality?)

why is x'' (acceleration) always in the direction if x increasing?

It looks as if you have this backwards. The acceleration (Force) is in the opposite direction as the sign of the displacement for SHM.

You will have to quote which book you refer to. You may be 'selecting' a particular passage which is confusing you. The basic equation of motion is clear enough and the book will certainly have got that right so it must be your interpretation of the wording in the book, if the wording seems to contradict the Maths.

 

[koliko987]

Acc is the same Acc, wherever you come across it. (How could it be otherwise if the Maths is used to model the reality?)It looks as if you have this backwards. The acceleration (Force) is in the opposite direction as the sign of the displacement for SHM.

You will have to quote which book you refer to. You may be 'selecting' a particular passage which is confusing you. The basic equation of motion is clear enough and the book will certainly have got that right so it must be your interpretation of the wording in the book, if the wording seems to contradict the Maths.

I attached a picture of two pages in the book:

The left pic is explaining the theory generally and the right is an example of proving SHM with a spring on a smooth horizontal table. They both have notes in yellow boxes saying x'' is in the direction of increasing x. Do you have an idea of what the book is trying to say? Thanks

 

[Delta2]

Book is right that the acceleration is always directed towards O. But then on the next statement (x'' is always in the direction of increasing x) i think it changes the semantics of the word "direction" and it means just the slope of the line, sorry my english fail me here, but in greek we use different words when we want to say "direction" or "towards to" but in english the word is the same.

 

[sophiecentaur]

I attached a picture of two pages in the book:

The left pic is explaining the theory generally and the right is an example of proving SHM with a spring on a smooth horizontal table. They both have notes in yellow boxes saying x'' is in the direction of increasing x. Do you have an idea of what the book is trying to say? Thanks

That phrase reads like complete gobbledegook. The Mathematical expression just beside it is, of course, correct. I guess the author had something in his head that suggested those words might help. They clearly don't. If in doubt about things like this, you should always stick with the Maths and look at another reference to check there is no error (equation layout / signs / etc.)
PS I wonder if he means that the acceleration acts 'along the line of x'.
It's a mystery - but you understand it so no harm done!