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I. Mechanics(related Past papers' questions and discussion)

By the end of this month - May, I am planning to complete the very first portion of Physics, Mechanics, out of eight different portions. Underneath this post, and in this thread, the solutions of questions of the past fifteen years related to Mechanics will be posed. The basic aim of this thread will be to have a discussion regarding the different facets of the subject-matter and to save the past papers' solutions.

[B]All those who are going to opt Physics as an optional subject will be welcomed for sharing the stuff related to the portion, Mechanics, shared below with the bullet "I."[/B]

[I][COLOR="DarkOrange"]NOTE: Please do not comment below if you have any unrelated thing to share.[/COLOR][/I] (Sorry for an orange colour, as red is reserved for moderators maybe; but I hope you will not skip this important note)

[IMG]http://i.imgur.com/mtpnwm4l.jpg?1[/IMG]

Which is the correct numerical in paper 2000

Find the unit vector perpendicular to i+j and j+?
Or
Find the unit vector perpendicular to i+j and j+k

As per my knowledge second option is right and its typing error in first one. Reply must so i could try it:thinking

Yes, it is "j+k."

Here is the solution:
[IMG]http://i.imgur.com/NDUYSnG.jpg[/IMG]

Redo this again, as I have just done it hurriedly. I hope it is correct; however, if you find any error, do mention it. :)

NOTE: I shall share another method as well, which involves matrices. That method seems more elegant to me than the aforementioned one.

Physics-Vectors(Same question as above)

Another method(Matrix method) just for confirmation.

[IMG]http://i.imgur.com/wG3ebK2.jpg[/IMG]

Error in uploading second image. Same method as solution 1 same answer. Finally i solved the vector problem without any help.:vic

[IMG]http://s23.postimg.org/avjp0yi7v/1432306718247.jpg[/IMG]

Calculate the angular momentum, about the Earth’s center of a 50 kg person on the equator of the rotating earth.

Check this. I am not sure

[IMG]http://s27.postimg.org/kn68ozmz7/20150522_210902.jpg[/IMG]

Angular momentum

Hey first of all, please use the vector notation where it is required. Otherwise, you will lose marks. Moreover, do not skip the pictorial representation of any question(where required).Those are the few things I noticed while checking your solution. Apart from that, you have correctly solved the whole problem. Also, do scrutinise my solution as well, for it will help us improve and revamp our concepts and solving methodology.

Here is my solution:
[IMG]http://i.imgur.com/JKKzdRB.jpg[/IMG]

Answers resemble but method is different. Your's is more authentic:bow

Check this also. If any other method, share.

[IMG]http://s9.postimg.org/j1s49rlkv/20150523_114733.jpg[/IMG]

Angular Momentum and Torque

That is how I solved:
[IMG]http://i.imgur.com/PVDiQEf.jpg[/IMG]
[IMG]http://i.imgur.com/CxyNb0p.jpg[/IMG]

A rod lies parallel to the x-axis of reference frame S, moving along this axis at a speed of 0.632 C. Its rest length is 1.68m. What will be its measured length in frame S?

[IMG]http://s7.postimg.org/qe6fm85q3/20150524_203734.jpg[/IMG]

.

Three vectors are given by:
a = 3i + 3j -2k
b = -i - 4j + 2k
c = 2i + 2j – k
Find (i) a. (b. × c), (ii) a. (b. + c), (iii) a. × (b + c)

The problem is written as (b. × c) it must be either (b × c or b . c). Am i right?

I have tried the first one through matrices and the simple cross method the answer is (b × c)= 3j + 6k
or it is b . c thats why i am getting the wrong answer. Resolve issues.

The second is

find the value of e so that A = ci – 2j + k and B = 2ci + cj – 4k are perpendicular.

Its not e its c. I solved it by dot product as

********* A . B = ABcos90°= 0 because A B are perpendicular. But didn,t get the answer.****

Regards.

[QUOTE=AZKAFATIMA;832226]Three vectors are given by:
a = 3i + 3j -2k
b = -i - 4j + 2k
c = 2i + 2j – k
Find (i) a. (b. × c), (ii) a. (b. + c), (iii) a. × (b + c)

The problem is written as (b. × c) it must be either (b × c or b . c). Am i right?

I have tried the first one through matrices and the simple cross method the answer is (b × c)= 3j + 6k
or it is b . c thats why i am getting the wrong answer. Resolve issues.

The second is

find the value of e so that A = ci – 2j + k and B = 2ci + cj – 4k are perpendicular.

Its not e its c. I solved it by dot product as

********* A . B = ABcos90°= 0 because A B are perpendicular. But didn,t get the answer.****

Regards.[/QUOTE]

It must be a.(b x c) . a.(b.c) is not possible.

I solved it as ABcos90 and got the same answer after applying quadratic formula. There is some mistake in the problem, for we get two values not one.:thinking

[QUOTE=AZKAFATIMA;832224]A rod lies parallel to the x-axis of reference frame S, moving along this axis at a speed of 0.632 C. Its rest length is 1.68m. What will be its measured length in frame S?

[IMG]http://s7.postimg.org/qe6fm85q3/20150524_203734.jpg[/IMG]

.[/QUOTE]

Yeah it is correct. One thing which made me a bit of uncomfortable was the naming convention. Lo (L not), instead of Lp should be used. Also give the percentage at the end which tells you how much the length of a rod contracts in moving frame.

Before formula, you should also mention "Using Lorentz–FitzGerald contraction, the length observed in relative motion with the object is,".

These are just my views; you may disagree.

If you are getting 2 answers then what is the confusion.Why you are thinking that there should be one answer. Both are correct and you can verify by putting the value of c and then taking the bot product.and your method is also correct if you are getting these 2 values.

Agree. I'll mention it in the remaining problems. Do you possess past papers of 2003, 2004 and 2006. These are not opening on the forum.

[QUOTE=AZKAFATIMA;832290]Agree. I'll mention it in the remaining problems. Do you possess past papers of 2003, 2004 and 2006. These are not opening on the forum.[/QUOTE]
No .

I verified and get the answer 0. I was in confusion about using quadratic formula in the vector portion.

To which part the problem show that Vr=r^n-2 r belong. I have not studied it.:thinking

Prove that thenecessary and sufficient condition for A B and C to be coplanar is A . B × C = 0

[IMG]http://s24.postimg.org/ognxejkxx/20150525_152837.jpg[/IMG]

[QUOTE=AZKAFATIMA;832367]To which part the problem show that Vr=r^n-2 r belong. I have not studied it.:thinking
[/QUOTE]

Here is the solution. I have skipped many steps, just to compel you to do it yourself. Lol It will make your concept clear.

[IMG]http://i.imgur.com/dFYnb4g.jpg[/IMG]

I have solved it as.

[IMG]http://s2.postimg.org/pzso4717t/20150525_190310.jpg[/IMG]

Sketch the function V = -yx+xy

[IMG]http://s16.postimg.org/79oo5s9jp/20150526_183025.jpg[/IMG]

Share the method if V is the scalar field

Can anybody elucidate divergence and curl. I studied but didn't get even a single notation.:cry
Should i go for video lectures as i am preparing myself.

[QUOTE=IslamabadKid;831787]Yes, it is "j+k."

Here is the solution:
[IMG]http://i.imgur.com/NDUYSnG.jpg[/IMG]

Redo this again, as I have just done it hurriedly. I hope it is correct; however, if you find any error, do mention it. :)

NOTE: I shall share another method as well, which involves matrices. That method seems more elegant to me than the aforementioned one.[/QUOTE]

Isn't the magnitude at the end wrong? Shouldn't it be "Underroot 3?"

Mechanics queries

[QUOTE=Noore776;836809]Can anybody elucidate divergence and curl. I studied but didn't get even a single notation.:cry
Should i go for video lectures as i am preparing myself.[/QUOTE]

I shall explain you both this morning[B] in detail.[/B] (2:59 AM right now lol)

[QUOTE=Norymberskie;836852]Isn't the magnitude at the end wrong? Shouldn't it be "Underroot 3?"[/QUOTE]

The magnitude is underroot 3! I left last step of finding unit vector, which has to be: Given vector - i-j+k - over its magnitude, for the member who asked this question.

AxB gives you a perpendicular vector, not perpendicular *unit* vector.

(1) doubletick at the end means you have everything now at your disposal to find the perpendicular unit(1) vector.

Hope this makes it clear!

[QUOTE=Noore776;836809]
Should i go for video lectures because i am preparing myself.[/QUOTE]

Video lectures is a good option but in this option we cannot ask any questions. You must seek guidance from the forum and if you belong to an educated family, you must take help from an engineer. Unfortunately, i have no engineer in my family.

Yes, i can avail this option. A brother in my family is studying civil engineering. Thanks for suggestion.

Vectors portion of Mechanics

[COLOR="MediumTurquoise"]Note: I could have explained it in a formal way, but I did not because it would have been hell boring and less interesting if I had done otherwise.[/COLOR]

[QUOTE=Noore776;836809]Can anybody elucidate divergence and curl.[B] I studied but didn't get even a single notation.[/B]:cry
Should i go for video lectures as i am preparing myself.[/QUOTE]

The clause in bold makes me adopt one of the two available options:
1) Should I directly explain you the curl and the divergence by disregarding the rudimentary stuff required to understand the above topics?

2) Should I elucidate the basic stuff first and then move on to the subject-matter? In short, should I spoon-feed you by elaborating the rudimentary stuff first?

Let flip the coin...

IslamabadKid vs IslamabadKid's intuition...

Ramez Raja: "Heads or Tails?"
IslamabadKid: "Heads"

Ramez Raja while tossing the coin in the air: "Heads is the call..."
*coin dancing in the air while singing: Main uddi uddi javan Hawa de nul... touches the ground... settles on the ground*

Ramez Raja: "It's tails!!! IslamabadKid's intuition, what do you want to choose?"

IslamabadKid's intuition: "I want choose the option 2!"

So it is settled! IslamabadKid succumbed to his intuition, and ready to explain the rudimentary stuff first.
--------------------------------------------------------------------------------
[B]The very first question that arises in one's mind is: What is so special about the vectors?[/B]

The answer to that is pretty straightforward, which is the [B]"vector"[/B] itself.

[B]The very next question that loomed up in one's mind is: Alright! Although that was obvious, what exactly is the vector?[/B]

Vector is the quantity that has a [B]magnitude [/B]as well as the [B]direction[/B].

[B]So you mean to say that anything that has some value associated to it and it is moving, heading or sets off in the specific path of its movement?
[/B]

Yes.

[B]Okay! If it is so, is there any [COLOR="DarkOrange"]general representation [/COLOR]of that? This question I am asking because whenever I travel, I see few symbols on specific points along the road, pointing towards the specific path. Also, "seedha aagay jayein... jaatay jayein... jaatay jayein... phir saj'jay ho jayein... aagay chaltay jayein... aur phir khabbay ho jayein... wahan aap ko aik Rickshaw nazar aye ga... uss Rickshay walay say sahi raasta pata karein, kyun kay mujay bhi nahi pata keh apki destination kahan hay exactly!"

That seedha, khab'ba, saj'ja are the specific directions, and the distance(s) we cover is/are the magnitude(s) associated to that path(s).[/B]

The most obnoxious, but true and to-the-point, interpretations you have mentioned. As far as the only question residing in that whole essay, which was how to represent the vector, is concerned, you need to imagine an arrow - [B]the real arrow[/B], like:

x------>

That "[B]x[/B]", which you stick in the rope of a bow, is the back of an arrow.
That "[B]>[/B]", which can penetrate into one's body easily because of its pointy surface, is the front of an arrow.

[I][B]Did you know:[/B] have you seen a real arrow in your life? If so, have you noticed one cool thing about it? If not, let me tell that.

If you look at the arrow from its front, you will see just a "dot" of the front pointy surface. If you look at the arrow from its back, you will only see the "cross"(x) of its rear end.

Whenever you see [B]encircled "."(dot)[/B] on the paper or a book, it means the arrow is coming out of the paper; and if you see [B]encircled "x"(cross)[/B] on the paper or a book, it means the arrow is going into the paper.

I hope it makes you remember the encircled "." and encircled "x" convection.[/I]
-----------------------------------------------------------------------------------

Now [B]imagine [/B] your room's [B]one wall[/B] out of four is firing vectors instead of bullets. *lol* The condition is that the vectors do not cross each other, meaning they are being fired in the straight lines like:

--> -----> --------> --------->
--> -----> --------> --------->
--> -----> --------> --------->
--> -----> --------> --------->
--> -----> --------> --------->
--> -----> --------> --------->
--> -----> --------> --------->
(Vectors* travelling along x-axis)
*Vector field(shall discuss later)

As you can see, the vectors that are being fired by [B]your left wall[/B] are heading along the x-axis, not y-axis or z-axis.

What if the vectors are fired by [B]the ground of your room[/B] in upward direction?
It means they(vectors) will be heading along the z-axis, not x-axis or y-axis.

What if vectors are fired by[B] the wall behind you[/B] in forward direction?
It means those vectors will be heading along the y-axis, not x-axis or z-axis.

What we have learned from above discussion is that the vector field* can be along any direction.

Now if you want to know of the[B] change in vector field(vectors in your room)[/B] happening [B]at any point[/B] in the room, all you need to do is to find the change along x-axis by ignoring y-axis and z-axis, to find the change along y-axis by ignoring x-axis and z-axis, and to find the change along z-axis by ignoring x-axis and y-axis.

Add all of them, you will get the result.

Now in mathematics/Physics, we use "partial fractions" so as to encounter that ignoring part of the above statements. What do I mean by that? It means the following:

d/dx i = change along x-axis by ignoring y-axis and z-axis.
d/dy j = change along y-axis by ignoring x-axis and z-axis.
d/dz k = change along z-axis by ignoring x-axis and y-axis.

[B]Please make sure that "d" is CURVY. I cannot type the curvy "d" in this forum as it does not support the mathematical symbols.

Also, i,j, and k represents the unit vectors along x, y and z axis respectively. Furthermore, there will be "caps" on top of i, j and k.[/B]

Now the [B]overall change [/B] of[B] vector field[/B],which is called the [B]GRADIENT[/B], can be found by adding all of the above notions, meaning:

d/dx i + d/dy j + d/dz k = Overall change of [B]vector field [/B]= Gradient of [B]vector field[/B]

Gradient is represented by the symbol "∇".

Please understand all of the above points first, and tell me what is unclear to you.

Once you understand all of the above discussion, we shall move on to the next, important topics:
1) Vector fields,
2) Divergence,
3) Curl.

All of the above topics especially (2) and (3) cannot be explained without the knowledge of gradients, vectors, and vector fields.

I shall use the aforementioned jargon over and over again while explaining Divergence and the Curl. Therefore, please understand that first.

Do let me know when you completely comprehend everything mentioned in this post!

If you face any problem, do mention that too.

P.S: 30-35% course of vectors is explained in this post. <--- just to motivate you!

*Suspense* lol (to-be-continued)

Yes, i got it in a humorous way. Sorry to say, i will only seek guidance in the most difficult topics. Did not want to disturb your pace because of me. Thanks

[QUOTE=Noore776;837762]Yes, i got it in a humorous way. Sorry to say, i will only seek guidance in the most difficult topics. Did not want to disturb your pace because of me. Thanks[/QUOTE]

First, I come here in my spare time.
Second, explaining something does not disturb my pace; it makes me revise my concepts.
Third, humorous way is one way to entertain oneself after a hectic day.
Fourth, getting something "economical" out of that humorous way is its another benefit.
Fifth, and above all, you cannot understand "curl and divergence" without the knowledge of gradients, as ∇.f = divergence; ∇xf = curl.

Regards.

Divergence and Curl - Explanation

[QUOTE=Noore776;837762]Yes, i got it in a humorous way. Sorry to say, i will only seek guidance in the most difficult topics. Did not want to disturb your pace because of me. Thanks[/QUOTE]

Continuation of the last post, of which theme was rallying around the following topics:
1) Vectors and their conventions;
2) Change in any direction;
3) Mathematical representation of that change - aka Gradients;
4) A slight introduction to the [B]vector field[/B].

Now let me continue from the topic I left off the discussion, which was the [B]vector field[/B].

The question is: What exactly is the vector field?

But hey wait! You know what a vector is, but what about the "field?"

In simple terminology, the word "field" can be defined as an area of an open land.

The very next question is: Okay! If that is the definition of a field, how would you connect its definition to the notion "vector field?"

The simple interpretation you can get by considering the definition of the field is that we have the area, which is filled with vectors.

The formal definition of the vector field is:

[QUOTE]A function of a space whose value at each point is a vector quantity.[/QUOTE]

Let us dissect the aforementioned definition!

[B]1) Space[/B]: You may call that the space is tantamount to your room's space. In-short, vacant area inside the four-walled room.

Let me relate it to the real world!

For instance, your parents have bought a new house, and now you are ready to transfer each and everything of yours into your room.

At first glance, you would say, "Oh! This is a big room! It has a lot of [B]space[/B] in its vicinity. Thank you Baba!"

*Now let's suppose that you have a weird -in others' frame of reference- habit of buying at least one book every week, meaning you have a lot of book that you need to accommodate into your room.*

Now you are "transferring" all of your stuff into your new room! After ten minutes, you will realise that you do not have [B]enough space[/B] in your room for your books. You reaction might be:*sigh* + "Awein he khush tha/the main kamra dekh k! Now I do not have [B]space[/B] for my books."

Now let me take "one" book and map it using Mathematics and Physics.

Every book has width, height, and depth, meaning it is just a 3-dimensional "mass" taking the space, which is width*height*depth(volume).

Why have I take 3-dimensions? because it encompasses the lower dimensions as well - which are 1-dimension and 2-dimensions; therefore, I do not want to go deep into the lower dimensions, for the intuition behind them is fairly easy. Later on, I shall take 2-dimensional area for ease.

[B]2) At each point[/B]: Meaning every point in the space; like:
[IMG]http://i.imgur.com/oTaGPQd.png[/IMG]

Suppose above is your room, and the red dots represents the "points" in the "space."

NOTE: There are infinite points; I have shown very few.

[B]3)Vector quantity[/B]: I have explained it in detail in my last post. I am assuming that you have understood that concept.

[COLOR="DarkOrange"]Now let us connect the dots, which in this case are three mentioned above[/COLOR];

Every point in the space is represented as a vector. Or the other way to elucidate is that a vector - a quantity that has a magnitude as well as direction- is assigned to each and every point in the space.

Like:
[IMG]http://i.imgur.com/8rUOPGN.png[/IMG]

Again, I have drawn very few vectors associated with the points in the space.

Having said that, the tragic story of vector field comes to its end.

It's ending -happy or sad- has been left to your imagination.
-------------------------------------------------------------------------------------

Let me show you few of the vector fields:

[IMG]http://mathinsight.org/media/applet/image/large/rotating_vector_field_3D.png[/IMG]

The above picture shows you the 3-dimensional "space" populated with the "vectors" assigned to few "points." You may relate it to your room, and imagine the vectors flying around you like mini-aeroplanes. You can imagine really cute scenario before going to sleep about 3-dimensional space and the vector field.

[IMG]http://mathinsight.org/media/image/image/vector_field_implosion.png[/IMG]

The above picture shows you the 2-dimensional "space" populated with the "vectors" assigned to few "points." You may relate it with A-4 paper, and imagine that vectors are moving on that paper.

y
^
|

---> ----> ----> ---> --> ->

->x

The above stream of vectors, imagine, are along x-axis only. <-- One dimensional vector field.

(The end)

Next topic: Divergence(Finally)! I hope you get something out of the post above regarding vector fields.

Cheerio!