Force, accleration vectors or not

[alkaspeltzar]

Taking high school physics, i have learned that force and acceleration are vectors. But i have noticed prior to this, we have never worried about them as such. Most of the problems simply treat them like any other measurment or value, ignoring direction. Almost like scalars.

So my question is, can we do this since for most problems it is just simpler and until we have more complexity with varying directions, we just work with force as purely 20lbs, 200 Newtons, ignoring the direction? Same with acceleration, we talk about it simply a some number of meter/second squared. Even the book will write its answers as"the force is 22.5lbs"...no direction.

Do people many times use the words force and acceleration as synomymous with the force or acceleration magnitude? Proabaly just the way it is

Thanks

 

[Doc Al]

So my question is, can we do this since for most problems it is just simpler and until we have more complexity with varying directions, we just work with force as purely 20lbs, 200 Newtons, ignoring the direction? Same with acceleration, we talk about it simply a some number of meter/second squared. Even the book will write its answers as"the force is 22.5lbs"...no direction.

Sometimes it just doesn't matter. Sometimes the direction is obvious. Sometimes all you need is the magnitude.

 

[Chestermiller]

When the direction of the velocity vector is changing along the trajectory, this translates into a component of acceleration (and net force) in the direction perpendicular to the velocity vector. The acceleration vector is equal to the rate of change of velocity vector along the trajectory. This is when it is important to include the directions of the vectors in the physical analysis.

 

[CWatters]

Most of the time you will need to treat them as vectors.

Relatively simple problems such as those involving balls thrown vertically upwards involve vectors but you may not have realized it - for example the initial velocity is in one direction (upwards) and the acceleration is acting in the oposite direction (downwards). You may not have realized that by assigning up or down as positive you are treating them as vectors.

 

[CWatters]

Do people many times use the words force and acceleration as synomymous with the force or acceleration magnitude?

Sometimes the direction is implied, for example gravity usually (but not allways) acts downwards. Best get into the habit of specifying the direction with your answers when possible.

 

[alkaspeltzar]

Sometimes the direction is implied, for example gravity usually (but not allways) acts downwards. Best get into the habit of specifying the direction with your answers when possible.

Okay CWatters, tell me if i have this right.

Acceleration and force strictly speaking are vector quantities, but many times since direction is implied or not important, we just simply work with them generally. This is why we often talk about the magnitude of force as a force, likewise with acceleration right?

But at the end of the day, we should try to include direction to be proper?

Would you agree?

 

[Chestermiller]

Okay CWatters, tell me if i have this right.

Acceleration and force strictly speaking are vector quantities, but many times since direction is implied or not important, we just simply work with them generally. This is why we often talk about the magnitude of force as a force, likewise with acceleration right?

But at the end of the day, we should try to include direction to be proper?

Would you agree?

It's much more than that. Consider the following:
Suppose you have a particle continually moving around the circumference of a circle at a constant speed. The particle keeps orbiting the center of the circle for all time. Here are some questions:

Is the particle accelerating? If you decide that the particle is accelerating, what is the direction of its acceleration vector?

Do you now see why it is important to include the directions of the velocity vector and the acceleration vector in your analysis?

 

[alkaspeltzar]

I get it is more than that. I am just wondering if my reasoning is correct as to why my textbook simplfies force and acceleration. That is all i am asking.

I agree, they should include direction but some many times my book leaves it as vague/understood.

 

[Chestermiller]

I get it is more than that. I am just wondering if my reasoning is correct as to why my textbook simplfies force and acceleration. That is all i am asking.

I agree, they should include direction but some many times my book leaves it as vague/understood.

Maybe your book isn't so hot, or maybe it's geared for the most elementary level. Is this a freshman physics course, or is it high school level? If it is freshman level, it should more clearly explain why including the directions of vectors is important, and present ample examples and problems to emphasize this.

Chet

 

[Doc Al]

I get it is more than that. I am just wondering if my reasoning is correct as to why my textbook simplfies force and acceleration. That is all i am asking.

I agree, they should include direction but some many times my book leaves it as vague/understood.

What might prove helpful to you is for you to post a few problems from your book (in the Intro Physics HW section, not here) along with your solutions. (Obviously, choose ones where you think there's an issue.) Then we can comment on why or why not the direction of the force was needed. (Sometimes, even when the direction is needed to solve the problem, all they ask for is the magnitude.)

 

[CWatters]

Okay CWatters, tell me if i have this right.

Acceleration and force strictly speaking are vector quantities, but many times since direction is implied or not important, we just simply work with them generally.

No. I would say most of the time you have to treat them as vectors. For example it's very common for velocity and acceleration to act in different directions.

I agree it would be interesting to see an example from your book. If you post it into the Intro Physics Homework section please post a link to it here.

 

[Travis_King]

That's why forces are described in systems like Cartesian coordinates.

 

[alkaspeltzar]

Examples from the Book

Okay, I have attached examples from the book. I know how to get to the answers, that is not what I want to know.

What is want to know is this:
1. In example 4.7, they begin by talking about the forces in the problems, as you see in bold letters with arrows denoting vector quantities. But as they progress, they simply look at the x and y components, which are magnitudes. They start talking about Ax as acceleration, but as far as I can tell, that is really only the magnitude of the acceleration in the x direction, so why do they call it acceleration?

2. In example 4.4, they just simply talk about acceleration and force without direction..why?

So those are my question. Why do they sometimes leave out directions as in ex. 4.4? And is it normal when breaking down the vectors, looking only at the components/magnitudes, to refer to them and call them Force, or tension.

Please review my example and read the remarks below. I guess I get confused because if it force is a vector, they should talk about it as one all the time and sometimes it isn't.

Thanks

 

[alkaspeltzar]

Here is another example, 4-2. They define the force vector at the beginning as F=Fxi, where Fx is the magnitude.

But then step #3 asks use to find the force(Fx). I agree that forces really should be described with both their strength/magnitude and direction, but many times we generally refer to the magnitude as just "force" too. At least that is what this book is doing?

Is this just an English shortcut the book is using that is confusing me? Maybe that is all it is, fudging the words..

Thanks for all the help. I appreciate the conversation.

 

[sophiecentaur]

But Fx DOES specify direction. i.e. the force in the x direction. Are you familiar with resolving vectors?

 

[alkaspeltzar]

Yes, I understand resolving vectors. And I get that Fx does show direction...but it is not a vector according to my book, only the scalar magnitude of the force along x-axis.

So that is where the confusion comes in. IS Fx a vector? If not, as the book says it is the component of F along the x-axis which is a scalar, why do they still refer to it as force?

They should say, find the magnitude of force along x-axis,,Fx instead of find the force Fx. I believe that would be less confusing

 

[sophiecentaur]

If you look in Example 4-2 you will see the actual vector defined as Fxi where i is a unit vector in the direction x. So Fx is a scalar. Having done that then they really don't need to spell it out all the time.
I think, by now, that you understand what it's all about but may still be annoyed at the way they're writing it down? It doesn't seem all that sloppy to me. They do put an arrow over F when they are referring to the vector force.

 

[alkaspeltzar]

I agree with that Sophiecentaur, the actual vector is Fxi, and when trying to find it, you need to calculate the magnitude of Force in the x-direction, using the equation fx=mAx. At that point they should say THAT, find the magnitude of force, fx, but instead, they don't need to spell it out. By saying find the force, referring to Fx you know what they mean.

I guess that was my question and you have answered it. Don't get to caught up on the words, sometimes we leave out details because it is already understood and doesn't need to be spelled out. That explains why in my other example, 4-4, they simply just called out force and acceleration without direction, because it was unnecessary in that problem. I am trying to be to perfect. Sorry for the confusion. THanks for the help

 

[Chestermiller]

It was very helpful to see your examples. I can now see where you are coming from.

If the motion is in a straight line, and/or the only relevant forces and accelerations depend on only one spatial variable, then it is not important to account for the vectorial nature of forces, velocities, and acceleration. However, if there exist forces and accelerations in more than one direction that need to be included in the analysis, then to solve your problem, you need to resolve the equations into components. Even though the force balance can usually be expressed exclusively in vectorial form (i.e., in terms of bold letters with arrows over them), when you solve a practical problem, you usually need to resolve the forces and accelerations into component form to get a mathematical solution to your problem. Think about trying to solve some of your problems without doing this.

Chet

 

[sophiecentaur]

I agree with that Sophiecentaur, the actual vector is Fxi, and when trying to find it, you need to calculate the magnitude of Force in the x-direction, using the equation fx=mAx. At that point they should say THAT, find the magnitude of force, fx, but instead, they don't need to spell it out. By saying find the force, referring to Fx you know what they mean.

I guess that was my question and you have answered it. Don't get to caught up on the words, sometimes we leave out details because it is already understood and doesn't need to be spelled out. That explains why in my other example, 4-4, they simply just called out force and acceleration without direction, because it was unnecessary in that problem. I am trying to be to perfect. Sorry for the confusion. THanks for the help

I can't see the contents of the pages previous to p95 but are you sure they don't mention the idea of unit vectors, in the directions of the xyz axes? If they do, then I can't see why you have had any problem. There is nothing in those examples that could be construed as confusing or even 'imperfect' if they have already told you what i stands for. If they haven't introduced unit vectors then they are being really really sloppy.
Life's too short and paper is too scarce for people not to be using well-known abbreviations. Where would you stop, if you qualified and defined everything before making a statement?

 

[alkaspeltzar]

Thank you for the help. I guess i was trying to be so perfect and you are right, sometime we shorthand otherwise the world would get wordy.

Chestermiller, thanks for you input too. That makes sense why many times, we don't have to account for the vectoral direction. Many of the problems i deal with simplify them down so we don't have too.

Glad you both were able to relate and understand my question. I thought i was going crazy for a minute. :)

 

[alkaspeltzar]

It was very helpful to see your examples. I can now see where you are coming from.

If the motion is in a straight line, and/or the only relevant forces and accelerations depend on only one spatial variable, then it is not important to account for the vectorial nature of forces, velocities, and acceleration. However, if there exist forces and accelerations in more than one direction that need to be included in the analysis, then to solve your problem, you need to resolve the equations into components. Even though the force balance can usually be expressed exclusively in vectorial form (i.e., in terms of bold letters with arrows over them), when you solve a practical problem, you usually need to resolve the forces and accelerations into component form to get a mathematical solution to your problem. Think about trying to solve some of your problems without doing this.

Chet

Chet, is that what we are really doing when we work with vectors in scalar notation/rectangular components? I see in my book we work with force as a vector, breaking it down into is vector components and freebody diagram. Then, using Fx=Fcostheta and Fy=Fsintheta, we find the scalar components that are really representing the component vectors. We many times work witht the values as negative or positive to reference a direction. Doing our calcuations like this is easier as you said.


I guess that is one question i have, are Fx and Fy sometimes negative to represent direction or what does it mean when they are negative?

 

[Doc Al]

I guess that is one question i have, are Fx and Fy sometimes negative to represent direction or what does it mean when they are negative?

That's all it means.

 

[alkaspeltzar]

So that explains it, we know forces are vectors, just many times it is easier to break them down into rectangular components where we can use regular algrebra. The components then are more less scalar values that we treat as the forces since they are representing them...correct?

Once solving the problem, we should work our solution back to specify the magnitufe and direction to properly describe the force. Does this sound right?

 

[Doc Al]

So that explains it, we know forces are vectors, just many times it is easier to break them down into rectangular components where we can use regular algrebra. The components then are more less scalar values that we treat as the forces since they are representing them...correct?

Sounds good. But note that each component is really a scalar value times a unit vector.

Once solving the problem, we should work our solution back to specify the magnitufe and direction to properly describe the force. Does this sound right?

Often that's what will be required. But realize that expressing a vector in terms of components or in terms of a magnitude in a direction are entirely equivalent.

 

[alkaspeltzar]

Doc Al, but according to my physics book, I can represent a vector either graphically, as a magnitude with direction, or in terms of vector components. Please see the attached Example 3-2

In it, it shows how to add the vectors graphically and then thru the components. It even states in the remarks that a vector can be expressed thru the components, which to me are just scalars right? Aren't Bx, By, Cx, Cy all just scalar components of the vector, no unit vector is included?

That bring up one more question, how can BX and BY etc be considered scalars, technically don't they include direction by the inclusion of the suffix "x or y" and negative values? Or are they scalar values that technically represent the direction thru the suffixes and neg/pos signs, therefore they are still scalars but can be used to express a vector.

 

[alkaspeltzar]

this is right out of my book, the definition of components, how they can be used to express a vector.

http://books.google.com/books?id=At...ectangular components are not vectors&f=false

My question is, how are the components not considered vectors. They are technically describing indirectly direction and magnitude, so what makes them different. Thank to the book, I know I can work with these scalars to express/work with vectors, I guess though I just don't see the big difference. Thanks

 

[Doc Al]

Doc Al, but according to my physics book, I can represent a vector either graphically, as a magnitude with direction, or in terms of vector components.

Which is why I said those representations are equivalent.

Please see the attached Example 3-2

In it, it shows how to add the vectors graphically and then thru the components. It even states in the remarks that a vector can be expressed thru the components, which to me are just scalars right? Aren't Bx, By, Cx, Cy all just scalar components of the vector, no unit vector is included?


That bring up one more question, how can BX and BY etc be considered scalars, technically don't they include direction by the inclusion of the suffix "x or y" and negative values? Or are they scalar values that technically represent the direction thru the suffixes and neg/pos signs, therefore they are still scalars but can be used to express a vector.

Bx, etc., are just numbers and thus scalars. Of course, the unit vector is implied.

 

[alkaspeltzar]

Doc Al, can you please tell me if this is right then, just so I have closure. I would really appreciate it:

So, I know for example that a force is a vector. But when working with forces, we many times find it helpful to draw a free-body diagram, showing the force and/or any of the vector components. Then to work with the vector, we could use polar notation, or unit vector notation, but many times for the sake of ease we use component notation, breaking it down into the Fx and Fy components. This many times is done using the equation Fx=Fcostheta and Fy=Fsintheta. It is helpful because now we can use regular algebra since Fx and Fy are scalars.

In the end we can express the vector in terms of Fx and Fy because even though they are scalars, direction is implied and it gives the same information as say having the magnitude and direction or having it in unit vector notation. That is what I am getting out of my book, that inorder to process vectors, we many times break them down into x-y force components, working with the vectors more/less as a scalars. Just like the examples showed, we could leave it expressed that way, it is the same. But sometimes we convert it back for a specific solution.

IS this right? I guess maybe I should not worry so much. I guess I have always done this and never realized it. I mainly would take a force/vector, break it down and analyze it simply as these scalars Fx, Fy, when really the whole time that was representing the vector components and doing the math.

 

[Doc Al]

Doc Al, can you please tell me if this is right then, just so I have closure. I would really appreciate it:

So, I know for example that a force is a vector. But when working with forces, we many times find it helpful to draw a free-body diagram, showing the force and/or any of the vector components. Then to work with the vector, we could use polar notation, or unit vector notation, but many times for the sake of ease we use component notation, breaking it down into the Fx and Fy components. This many times is done using the equation Fx=Fcostheta and Fy=Fsintheta. It is helpful because now we can use regular algebra since Fx and Fy are scalars.

In the end we can express the vector in terms of Fx and Fy because even though they are scalars, direction is implied and it gives the same information as say having the magnitude and direction or having it in unit vector notation. That is what I am getting out of my book, that inorder to process vectors, we many times break them down into x-y force components, working with the vectors more/less as a scalars. Just like the examples showed, we could leave it expressed that way, it is the same. But sometimes we convert it back for a specific solution.

IS this right? I guess maybe I should not worry so much. I guess I have always done this and never realized it. I mainly would take a force/vector, break it down and analyze it simply as these scalars Fx, Fy, when really the whole time that was representing the vector components and doing the math.

I'd say you are right. And I agree that you shouldn't worry so much and get hung up on semantics.

The real thing to worry about is: Can you solve the problems?

 

[alkaspeltzar]

Your right, I can do the problems. I know what a force is, how vectors have both magnitude and direction, and I know that we can break them down via rectangular components in math to make working with them easier. In the end, that is right, it works and follows the book, and I get my answers.

I know I get hung up on things, I guess it is just important that I know how to do it properly. I am not trying to be a physicist, but I would still like the understanding.

thanks for all the help again. Your patience has helped put my mind at ease

 

[alkaspeltzar]

Which is why I said those representations are equivalent.


Bx, etc., are just numbers and thus scalars. Of course, the unit vector is implied.

Doc Al, then why don't we call Bx, By etc in these examples vectors? Aren't they kinda representing them, so at the end of the day, what is the difference?

I guess one thing i am misunderstanding is how we break down vectors into their component notation, in terms of Fx and Fy, which is the scalar components. This is an equal representation of a vector, but if Fx already implies a direction by having "sub x", i don't see the distinction.

 

[alkaspeltzar]

Take a look at the scan out of my book. IT talks about the Scalar components and Vector components.

I understand we break the force into 2 vector components, then for the sake of the alegbra work with the scalar components. But when i look at a scalar componenet, say Fx =-500N, to me that means backwards in the X direction 500 Newtons, so how is that different from the vector Fx=-500Ni?

 

[CWatters]

Sorry to butt in but...

many times for the sake of ease we use component notation, breaking it down into the Fx and Fy components. This many times is done using the equation Fx=Fcostheta and Fy=Fsintheta. It is helpful because now we can use regular algebra since Fx and Fy are scalars.

Fx and Fy are still vectors.

The reason for spliting the original vector into components Fx and Fy is usually because you already have other vectors with components acting in the directions x and y. That allows you to take advantage of the fact that two vectors pointing in the same direction can easily be added together. The same applies to vectors that are the components of other vectors.

For example suppose you have numerous forces acting in random directions on a body. One way to work out the total force is to convert all the vectors to their components in some co-ordinate system (it could be x,y,z or some problem specific system, whatever is most convenient). Then add all the components and convert the result back to whatever form is required.

Just because the components are in the x,y,z directions doesn't make them scalars but it might allow them to be added as if they were scalars.

In some problems this process is best done in two or more stages. eg some vectors are broken down into components in say the i,j,k co-ordinate system and added together. The result is then converted to components in the x,y,z coordinate system where the components of other vectors are added. The result might need to be converted to a third co-ordinate system depending on what the problem requires.

 

[Doc Al]

Take a look at the scan out of my book. IT talks about the Scalar components and Vector components.

I understand we break the force into 2 vector components, then for the sake of the alegbra work with the scalar components. But when i look at a scalar componenet, say Fx =-500N, to me that means backwards in the X direction 500 Newtons, so how is that different from the vector Fx=-500Ni?

I agree with your book. I'd call Fx a scalar, as it's just a number; the vector form would be Fx i. But that's just a technicality. Of course we are talking about representing vectors using their components. (By writing Fx in bold, as Fx, you are merely creating your own vector notation. Thus Fx = Fx i.)

Scalar: 500N
Vector: 500N i

I really think you are making a big deal out of a triviality. Who really cares whether you call Fx a scalar or a vector, as long as you know how to represent a vector in terms of its components and perform all needed operations correctly.