## Friday, 24 October 2014

### Pulleys

A pulley is a simple mechanical device used either to change the direction of force or gain Mechanical Advantage or both. Questions from this topic, are usually combined with other topics in JEE and to solve such problems a basic knowledge of this topic is essential.

The Basic Concepts that you need to know are:

1. The pulley is considered to be massless and frictionless.
2. The string is masless and inextensible.
3. Tension is the same throughout the string.
4. The direction of tension force is along the string, away from the point of attachment.
Note: The first three points are valid only in Ideal Situations.

Using these concepts and Newton's Laws of Motions any problem related to pulleys can be solved. The following examples will illustrate how.

Eg. 1.  A man is pulling the loose end of the rope with a force F. What is the Force F required to hold the object of mass 10 kg at rest ? Also if the object goes up by 1 metre, by what distance should the man's hand go down?

Step 1 : Mark T1 and T2 on both strings with proper direcn at all points of contact. To save time ( and keep things neat ) I have marked tensions only at the contact point with the block and movable pulleys. The fixed ones don't really matter, all they are doing is changing the dircn of the force.
Remember : The direction of tension force is along the string, away from the point of attachment.

In actual problem solving, once you've practised enough you should be able to see directly that T2=2T1 and T3=4T1. Sp you can directly write it all in terms of T1.
Writing everything is a longer process and is only recommended for beginners.

Step 2 : Consider unknown accelerations of the movable pulleys and the block/blocks and assign them directions. ( If you assigned the wrong direction your answer will come out negative so don't worry ).

Note : Pulleys P1 and P2 are acted upon by identical forces and are placed identically with respect to the ropes. Therefore they will both have the same magnitude of acceleration.

Step 3 : Write the force equations for the movable pulleys and blocks. In case of Ideal Pulleys, the net force on a pulley should be equal to zero. For blocks, net force = ma

F=T1
P1 : 2T1 = T2
Note : P1 and P2 will give the same equation.
P3 : 2T2 = T3 = 4T1

For the Block :  T3+T1=5T1=ma1+mg
Note : Only these equations are enough to solve the first part of the question. Just substitute m=10kg and a1=0.

Step 4 : Relate different accelerations utilizing the fact that the length of the string is constant. This is the most confusing part. The concept sounds quite simple but it is difficult to apply. To get these equations draw each part of a string one beside the other, sort of like a barcode ( but with a lot more spacing ).

eg. Consider this simple diagram.

Each part of the string has been drawn differently. Also each end of a string has an acceleration equal to that of the object it is connected with. I have used different coloured accelerations for up and down. Decide on a sign convention, Since we have more downs here, let us take down as positive and up as negative. Now write down the accelerations with appropriate signs. Write the accelerations of the top end of the string on the LHS and the lower end of the string on the RHS with appropriate sign.
You will end up getting :

2a2 + 2a3=a1 - a4

In any question you have to do this for each string present.

Now try this method yourself to on the question we were solving. You should be getting :
a4=4a2+a1
a2=a3
a3=a1 ( The block can have only one acceleration ( Ideal case assumption, no torque on the block ) ).
On solving you will get a4=5a1
Velocity and displacements are also proportional to the acceleration. Therefore if the block moves up by 1 metre, the man's hand should pull the rope down by 5 metres.

This is all there is to know about pulleys. Do try more problems till you understand the concepts fully. If you don't understand something or if you find any mistake scroll down and comment.

The First Image is By César Rincón (Own work) [GFDL (http://www.gnu.org/copyleft/fdl.html) or CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/)], via Wikimedia Commons.
The rest of the Images are provided by me © Mridul Kothari

1. Thanks a lot for posting.You explain the things quite well.

1. Thank You :-)

2. What are the prerequisites to learn the above article?I am not able to understand it properly.

1. Can you tell which part you couldn't understand? A basic knowledge of Newton's Laws , and writing Force Equations from Free Body Diagrams is essential.

2. If a1=0,then all the tensions are zero and F=0.Then how come the block will be at rest?

3. I'm really sorry for that. It was an error on my part. It should have been ma1 + mg. Thank You for pointing it out.

4. In a4=4a2+a1,shouldn't a4 be with negative sign?
Also,how is a2=a3?

5. No there won't be a negative sign according to the directions I've considered in the picture.

For a2=a3, consider the second rope. You can break that one into two different ropes. In both of those sections, the upper part has an acceleration a2 upwards and the lower an acceleration a3 upwards. So if you write the constraint equations you will get a2=a3.

Just draw all sections of the rope, mark the accelerations correctly at each end and derive a relation, remembering that the length of the rope is constant.
If length of one part decreases at an accleration a' then the length of the other part should increase at the same acceleration.

Try it one more time and if you still have a problem understanding you can ask again :-).

3. By second rope,you mean that rope with tension T2,right?Since its upper part is connected to p1,it has an acceleration a2 up,and since its down part is connected to p3 it has acce. a3 upwards.So,we arrive at the relation this way:
2a2=2a3
a2=a3
Am i right?

1. Yes, Correct :-)

2. How did u arrive at the last statement after a4=5a1?

3. Suppose you start with the displacement equation. Consider that the 10 kg block moved up by a distance x1. Then draw all parts of the string and mark displacements at different points. Through a similar procedure you would obtain x4=5 x1.
Now if you differentiate this equation once you will get the velocity equation and then the acceleration equation.

The concept remains simple : The Length of String is Constant.
But so much can be derived from it.

4. Thanks for the quick replies.Now,i understood the above article.Hope u are not fed up of my questions.i wanted to ask u a few questions:
did u take CS in 12th std.?
do u answer any conceptual doubts?

1. Yes I took CS, And I can answer doubts ( I hope I can). But I don't know much about C++, I studied Java in class 12. But still Conceptual doubts should be solvable :-)

5. On what topic did u perform project?
Conceptual doubts not regarding CS but PCM.do u answer them?

1. Yeah any doubt is Ok.
My CS project was a school report card. Enter, save and edit marks of multiple students and calculate ranks, plus display it all effectively.

2. I am a Jee 2015 aspirant.At present stage,my situation is not good.I brought and kept big books in my cupboard but at present am not able to complete a single book.I know tidbits of all the topics but none completely.I wanted to ask u a question:
When do u say that a topic is completed?
What would u suggest to me?
Would u tell me the thermodynamics(chemistry) chapter in an understandable way?Right from when enthalpy starts.Because of that,am not able to understand equilibrium and electrochemistry.I referred 2-3 books for that but all are bouncers.Thanks for the reply.

3. Ok I will Try Thermodynamimcs Over the weekend. I would say a topic is complete after you've practiced some questions apart from studying theory. Only theory is not enough to complete a chapter. Practice questions too.
Since you know a bit of most of the chapters, you should probably start practicing questions. I was at a similar situation in Conics. At that time, I started studying chapters only through questions, no theory reading. I would get stuck an awful lot of times and when you get stuck refer to the solution. After doing 10-20 questions, you will have a basic idea of the concepts and formulas involved in that chapter and if you then read the theory your understanding will be further strengthened.
I used this approach for Conics ( that was my weak area ) and it was helpful.

6. Hi. I'm preparing for JEE15.I'm confident of my preparation in maths, except in PnC, Probability. In physics I'm good at Optics,Electromagnetism,Modern Physics and Mechanics I (till rotation). In chemistry I've done Physical and Inorganic properly but I'm very bad at Organic. How do you think I should go about my prep now?

1. Good that you are confident about most of the topics. Give more time to organic from now. Try and finish it in the next two months. You can't afford to leave Organic completely, that would be very stupid. Since you find it tough, start at the basics. Start with Stereo-chemistry and then the basic reactions. Also you can afford to leave bio-molecules if it really irritates you, but most of the other things are important. There's no substitute to hard work! Start at the basics, and practice questions alongside, practicing questions will help to recognize the exceptions, in Chemistry exceptions are very important.

http://www.masterorganicchemistry.com/2010/12/22/on-cats-part-7-the-fischer-projection/

2. What about physics? I have absolutely no clue about waves and thermo. Whatever thermo questions I solve in physics , I do it on the basis of what I learnt in chemistry(with correct conventions ofc ) and waves I havent even been taught (kind of a weird situation, i left my 11th coaching mid nov and joined my current coaching in maths, so i missed some topics. i covered what i had missed in maths and chem with self study but topics i missed in physics i never even read them once)

3. If you know thermodynamics in Chemistry you shouldn't really have a problem with Physics! There's less in Physics ( No Enthalpy, Free Energy or Entropy to deal with ). Go through it once, since you already know some thermodynamics it shouldn't take a lot of time.
Waves seem tough but it actually isn't. There are a few simple formulas which can be understood ( The ones related to harmonics and overtones ) and a few formulas which you just have to mug up ( Energy of the wave and loudness of sound ). Also make sure you learn up Doppler's effect. Doppler's effect and harmonics/overtones is the important part the rest can be mugged up.
In case of electromagnetic waves, there is surely lots to cover but you have study it for your boards either way so it shouldn't be that big a problem.

7. Hi..I'm an aspirant for jee 15. How do you suggest I can best increase my speed? Because, while answering tests at my coaching center, whenever I try to attempt more questions by increasing speed, I almost always end up making silly mistakes and hence end up getting even lower scores than when I attempt lesser questions. How do you suggest I can increase my speed without compromising on accuracy?

1. There's no shortcut. Practice, practice and keep practicing. And yes, you won't get any immediate improvements, it will take time, have faith and patience, and keep practicing. Over a few months you will definitely notice an improve in your marks at coaching.

8. 1. Yes, I scored close to 370.

2. Is it tougher than jee mains?
Also,did u prepare for english and logical reasoning?
My current situation is:
Physics:
Strong in mechanics,waves and thermo,electrostatics and current electricity,Optics
Weak in Magnetism and electromagnetic induction(am not able to understand them) and modern physics not yet completed
Maths:
Strong in algebra,Calculus,Trig,
Very weak in 2D Geometry and vectors and 3D geometry not completed
Chemistry:
All is studied in bits.Nothing is stronger and nothing is weaker.
I came to know that one should cover syllabus 6 months prior to the exam as most of the students do it.Is this true?I am completely pressurized because of this and haven't studied anything for the last four days just because of pressure.Am I in a good position compared to other aspirants or should I hurry to complete the syllabus?When did you complete your syllabus?

3. The questions are easier in BITSAT, but they are easy not just for you but for everyone. Overall quite tough to crack BITSAT.
If you are from ISC board, English shouldn't be a problem and Logical Reasoning isn't all that tough. It's much better to concentrate and improve your score in PCM.
Realise two things, first, you are lagging a bit since some of the topics that you don't know are part of your board syllabus too. Second getting pressurised and doing nothing is only going to make things much worse. Don't think too much, at this stage just keep studying as much as you can. Don't procrastinate or plan to drop. Give it your best shot. Make a study plan and follow it.
I too couldn't complete my syllabus on time ( I completed it in January) and I too wasted a hell lot of time by being scared and worried. I know how you feel, and I know it's hard to overcome that feeling ( I couldn't overcome it myself, hardly studied anything towards the end), but if you can somehow overcome it, you can be a winner. Watch this :
http://youtu.be/mgmVOuLgFB0

9. Let's assume the system actually works with an odd shaped weight(regular even weight as drawn will cause block to turn clockwise, another issue entirely). Is there any reliable way to predict the behavior of P1/P2/P3 when you pull the rope assuming a friction-less environment? All we can reliably say is P3 rise 1 meter and between P1 and P2, a net of 2 meter is risen. But this can be 0+2, 1+1, 2+0 or *insert your deity here* forbid -1+3. Will it be affected by the inertia of the rope/pulley(note i said friction-less not inertia-less)?

10. How did you get a4=4a2+a1 ?
I could not get it.
11. 12. 