Physics Help

[thewisecrab]

I have a few doubts in Physics, mainly in Vectors and Scalars. I asked my professor for clarifications, but I ended up more confused.

So here you go:

(please note: this was my first "advanced" physics lecture, since i finished the 10th grade, so I may have some really basic doubts that do need clarifications)

 

1) How do I resolve vectors into X and Y components? (for applying component rule of addition)

 

2) What is a unit vector?

 

3)I did not understand this relation at all:

x = i ^ ( the "^" is meant to go above "i")

y = j^ (same as above)

z = k ^ (same as above)

 

Can you'll please help me out? If I dont understand these concepts, my next lecture is will be completely out of sorts as it is based on this lecture.

[swansont]

I have a few doubts in Physics, mainly in Vectors and Scalars. I asked my professor for clarifications, but I ended up more confused.

So here you go:

(please note: this was my first "advanced" physics lecture, since i finished the 10th grade, so I may have some really basic doubts that do need clarifications)

 

1) How do I resolve vectors into X and Y components? (for applying component rule of addition)

 

2) What is a unit vector?

 

3)I did not understand this relation at all:

x = i ^ ( the "^" is meant to go above "i")

y = j^ (same as above)

z = k ^ (same as above)

 

Can you'll please help me out? If I dont understand these concepts, my next lecture is will be completely out of sorts as it is based on this lecture.

 

For 1) you have to apply a little bit of trig. You draw your vector in whatever coordinate system you are using, and make a right triangle with the vector and one of the axes. If you use the x-axis, the x-component will be the vector multiplied by cos(theta) and the y-component will be the vector multiplied by sin(theta). Theta is the angle between the vector and the x-axis.

 

If trigonometry gives you problems, you need to brush up on it.

 

A unit vector has length of 1 in a particular direction. That way you can represent another vector as the length (a scaler) multiplied by the unit vector.

 

i, j and k are unit vectors in the directions of the Cartesian axes, x, y and z.

[thewisecrab]

Understood Q)2 and Q1)

didnt understand Answer to Q3

And finally..Why is resolving the vectors done????

 

 

But thanks for a reply

Edited July 19, 2008 by thewisecrab

[Bignose]

Why is resolving the vectors done????

 

This is done primarily to make the math easier. Nature itself does not really care about what coordinate system/set of basis vector we use to describe something with. In fact, to be considered a "good" physical law, it has to be completely valid in all coordinate systems.

 

Let me give you an example of why the resolving is done.

 

Consider two identical cannonballs. You're atop a tower defending your castle from invaders, and you fire your cannon straight out horizontally. (so that the cannonball from the gun initially only has horizontal velocity.) Further say that since you forgot to call out "fire in the hole" when you shot your cannon off, you startled the guy manning the cannon next to yours, causing him to drop his cannonball he was trying to put into the end of his cannon. So, both cannonballs are let go at the exact same time, the only difference between them is that one has no horizontal velocity (the dropped one) and the other has some horizontal velocity (the fired one). Now, further assuming that the ground up to the tower is perfectly flat, which cannonball hits the ground first?

 

Obviously, you can use physics equations to determine the position, the velocity, the acceleration, etc. of the cannonballs are any time. But, the really neat thing is that by choosing a coordinate system that has one coordinate in the direction of gravity and another perpendicular to gravity along the line of the fired cannon, you can write separate equations for how far away from the tower the cannon ball is, and how high off the ground the cannon ball is. That is to say, with a good choice of coordinate systems, you can make those two quantities completely independent of each other. The distance away from the tower has no effect on the ball's height above the ground and vice versa. In this way, you should find that the equations for the height above the ground for the two balls is exactly the same. Only differences in the two ball's vertical components change the ball's equation for height above the ground. In this case, both vertical velocities were zero, even though they had very different horizontal velocities, so both vertical equations are the same.

 

This is the advantage of breaking vectors into components. You can quickly identify what affects what parts of the velocity.

 

Choice of coordinate system is usually done to make the problem as simple to answer as possible. For example, consider the flow of fluid in a tube. Almost always, the cylindrical coordinate system is used. This is a coordinate system that instead of the common x,y,z (also known as the Cartesian coordinate system) used r, [math]\theta[/math] ("theta") and z. r is the distance from the center of the pipe and [math]\theta[/math] is the angle around the pipe (like the hands on a clock face. The minute hand at 2:15 makes a different angle with the 12 than it does at 2:18 that angle is [math]\theta[/math]). z is the same. The reason this coordinate system is chosen is because it makes the math much, much easier. The governing equations of fluid mechanics are still perfectly valid if you wanted to use the x,y,z coordinates, but they are much harder to solve.

 

So, in the two examples here, the common theme is picking coordinates that make the problem easier to solve. In the first, we picked a coordinate to be in the direction of gravity. In the second, we picked a coordinate to go in the direction of the flow. In the first, we could have picked a coordinate direction that was 45 degrees to gravity. It wouldn't have been as easy to solve, but it can be done.

 

I hope that the ideas I've tried to present are clear. If not, please don't hesitate to ask more questions. But the main points are that using vectors to separate the components makes things easier. It may not seem so at first, but in the long run, it is the much preferred method because it is significantly easier.

[thewisecrab]

Thanks Man

Got It

One last query

During my classes (while doing numericals), the i and j units are used

then where is the k unit used? and what is the Z axis? is it used in 3D figures?

PS How did you write "Theta" as a symbol?

[swansont]

It's hard to draw three axes on a two-dimensional surface, so many problems are limited to x and y.

 

math (latex) tags:

[ math] \theta [ /math] will appear as [math] \theta [/math] if you omit the space after the [ (you can see the code for the symbol if you hover your mouse over it)