Solution to Probability

Let us consider only the 1st 25 places. Convince yourself that the people beyond position 25 in the queue do not change our probability of winning a free drink.

For these 25 people, there are 365^25 different birthday assignments.

Now, the only 'birthday assignments' where we win a prize are where the following 2 conditions hold:

1)The 1st 24 people have distinct birthdays: (365,24)*24! ways
2)Our birthday is 1 of these 24: (24,1) ways

So, favorable birthday assignments are (365,24)*24!*24.

Solution to Changing Angles

Lets write xSin(Q)=R. (call theta 'Q').

Differentiate wrt. 't' we get:
xCos(Q)(dQ/dt)+SinQ(dx/dt)=0.

Replace dx/dt by (v1+v2) to get the answer.

Changing Angles

Consider the setup depicted.
The hemisphere, of radius R, moves to the right with a speed of v2, and the lower end of a rod, long enough such that it leans on the hemisphere, moves to the left with a speed of v1.

Find the rate at which angle θ changes.
Courtesy: http://dhwanit-itstrue.blogspot.com/2011/12/problemhemisphere-and-rod.html

Change of Medium

I came to know that people couldn't get much out of my last post about 'Anti-reflectivity,' simply 'coz they didn't know about the co-efficients of reflectivity and transmissivity. So here it is.

   Consider a transverse "sinusoidal" wave travelling from a thin string to a thick one, across a 'joint'. Part of the wave is transmitted across the joint to the heavier string, part of it is reflected back to the thin one. Let the amplitude of the incident wave be A_i, and those of the reflected and transmitted waves be A_r and A_t respectively.
   If the wave travels in the first string with a velocity v₁, and in the second with velocity v₂, give me the coefficient of reflectivity, i.e. A_r/A_i, and the coefficient of transmissivity, A_t/A_i.

P.S.: The reflected and transmitted waves could also undergo a phase change, which is depicted mathematically by r or t being negative.

Hint: Use the fact that the "joint" exists. That is to say that the two ends cannot be at two different points. Also use the fact, that an 'acute' bend in a stretched string can only be accounted for by infinite tension!
By the way, the tension in both strings is the same.. isn't it?

P.P.S.: Though conservation of energy is a consequence, if you're struck, you may want to use it. (:

Probability

There is a big line of people waiting outside a bar for buying drinks. The owner comes out and says that the first person to have a birthday same as someone standing before him in the line gets a free drink.

You're standing at position 25 in the line. What is the probability that you get the free drink?

Note: no fancy probability tricks required. (Favorable cases)/(Total cases) will work.

Solution to Robot

As promised, here is a solution.

For N=15 and 9 turns, lets take a right step first. Then the path definitely looks like the one shown in the figure. Every tuple {(x1,x2,x3,x4,x5),(y1,y2,y3,y4,y5)} corresponds to a different path. Consider the equations:

x1+x2+x3+x4+x5=14 (xi>=1)-------(i)
y1+y2+y3+y4+y5=14 (yi>=1)-------(ii)

Let P be the number of solutions to (i) and Q be the number of solutions to (ii). Then the answer is 2*P*Q. The factor 2 comes from the fact that for every solution, there is a similar path where we initially take a down step.

Anti-Reflectivity

When light of intensity I reflects from a surface separating two media with refractive index n₁ and n₂, the intensity of the reflected light is
^(n₂-n₁)2/_(n2+n1)² .
To make reflection zero a thin layer of a material of refractive index n of thickness t is inserted between the two media. Give me the value of n and t such that light of wavelength λ is not reflected at all.

Robot

A robot moves from cell (1,1) to the cell (N,N). It has only 2 possible moves: right or down. A kink in the path is called a turn. For example, the path in the figure has 5 turns. The robot can make only 'k' turns.

Find the number of possible paths for:
N=15, K=9.

Example: for N=4 and k=2, the robot needs to go from (1,1) to (4,4) and make only 2 turns. There are 4 possible paths: RRDDDR, RDDDRR, DRRRDD and DDRRRD.

More on the center of rotation method:

Well, this sounds like I'm making it up, but the day I noticed the method, I was up all night trying to find out "cases" where the method won't work.

Consider the diagram (i). Try calculating the angular acc. of the main pulley using 2 methods: C.O.R. method, and the usual (and correct method). You will notice that C.O.R. method gives a wrong answer.

So, I concluded: apply C.O.R. method only for problems with single rigid bodies, with a center of rotation.

Now that you know the C.O.R. method, can you solve:this question that was specially designed to illustrate the C.O.R. method?

And now try these

Ïf you read the last 2 posts, here are some practice problems. Try writing down the expression for angular acceleration and then verify with check with the expressions in red.

Link to problems:
http://www.anujkalia.blogspot.com/2010/03/its-yours.html

Read the previous post first

Here's another example

In this question, you need to find out the angular acceleration of the pulley.
I'll use torque=moment of inertia*angular acceleration in a differente' fashion.

Write the torque about 'P' (the center of rotation is ALWAYS denoted by the letter 'P'):
Torque=m1*g*R-m2*g*R

Now whats the moment of inertia about 'P'? The rigid body contributes M*(K^2). But how much MOI do the masses 'm1' and 'm2' contribute? m1 contributes m1*(R^2). m2 contributes m2*(R^2).

Now, we write:
m1*g*R-m2*g*R=(M*(K^2)+ m1*(R^2)+m2*(R^2))*alpha.

Now find alpha.

Lets learn something new

Here's how I use center of rotation:

In this question, you need to find out the angular acceleration of the circular body.
I'll use torque=moment of inertia*angular acceleration in a differente' fashion.

Write the torque about 'P' (the center of rotation is ALWAYS denoted by the letter 'P'):
Torque=m*g*(2R)

Now whats the moment of inertia about 'P'? The rigid body contributes M*(K^2+R^2). But how much MOI does the mass 'm' contribute? It contributes m*(2R)^2. Note that 2R is the distance of the point 'T' from 'P'. i.e. I'm assuming 'm' to be at 'T'. Why? I don't know why but this ALWAYS works.

Now, we write:
m*g*(2R)=(M*(K^2+R^2)+ m*(2R)^2)*alpha.

Now find alpha.

Remember small things

A long wire of radius ‘a’ is carrying a direct current I.

From its surface at point A, an electron of charge −e (e > 0) escapes with velocity v₀ perpendicular to this surface.

(see Figure)

Analyze the motion to check whether the electron
- escapes to x -> ꝏ, or
- approaches some x -> x₀ (find it if yes), or
- comes back, after reaching an xmax (find it if yes) .

Ignore gravity.

< Courtesy: INPhO 2011 >

How far down the rabbit hole are we?

Well, I know this is gonna be a little boring here.. but bear with me..

Coz' I can guarantee, that if you bear with me, and you allow yourself to THINK and WONDER, you're gonna be surprised at the end.

What happens, when bits of matter, say marbles, are shot at a screen through a fine slit?

How does the pattern change, if we add another slit of the same thickness?

Now let's look at waves.

What pattern (I don't wanna know what you 'call' that pattern, but what it actually is) can you see on the screen if water ripples through a single slit?

( You may assume that a stone is dropped in water in front of a slit half submerged in water. )

Again tell me, what changes, if I add a second slit.

So now, we know what happens when matter passes through 2 slits, and what happens when waves pass through 2 slits. (Don't we?)

Let's go again.

What do we expect to see if electrons (tiny tiny bits of matter) are shot at a screen through a single slit?

And what if I add a second slit??

THINK.

If you think you have the answers, click here.

As of discussing together.. well, we can do that later, but first I want you people to answer these on your own.

Let's make this simpler.. assume that the slits are thin enough, not to permit more than one particles side-by-side.
And yes, our screens are 'intelligent', they record all intensity patterns. (:

Fear Factor

A spool of radius 100m and mass 100kg has 100 grooves of radii 1m, 2m, ..., 100m. We use pulleys to suspend masses from the grooves as shown. The mass (i)kg is associated with the groove of radius (100-i)m.

Find the angular acceleration of the spool.

So you don't know how to find the Radius of curvature?

Lets try it again.

A particle is projected with a velocity 10m/s at an angle 37* with horizontal. Whats the radius of curvature at t=0? At what instant is the radius of curvature minimum?

Find the radius as a function of time.

Solution: Radius of curvature

F=1i`+2j`+3k`
V=-2i`+3j`-k`

Now, the component of force perpendicular to the velocity is (m*|V|^2)/r.

This component is |F|*Sin(Q)=2*14/r
or sqrt(14)*Sin(Q)=(2*14)/r

or r=(2*sqrt(14))/Sin(Q)

Q is the angle b/w the 2 vectors F and V.

Radius of Curvature

Do you know how we find the radius of curvature of a particle's path?

A force 1i`+2j`+3k` acts on a particle (mass=2) moving with a velocity -2i`+3j`-k`. What is the radius of curvature of the particle's path at this instant?

Angular Momentum

A particle is projected with speed 'v' at an angle 'Q' with the horizontal. Find its angular momentum about the point of projection after time 't'.

7 elastic beads for the Dwarf-lords, in their halls of stone

Seven identical beads are sliding on a meter rod as shown. Their positions and velocities are listed. A positive velocity indicates motion to the right.

When a bead slides off the meter rod, a Dragon awakes. Presently, its t=0.

At what times does a Dragon awake? Note that a total of 7 Dragons will wake up. List all the 7 time instants.

Click the image to enlarge and notice the minus signs.

Integral Calculus

Let f : R -> (0,∞) be a decreasing function. For n ∈ N.

We define,     a(n) = f(1) + f(2) + ... +f(n) - ₁ ∫ ⁿ f .
Show that the infinite sequence (a(n)) is a bounded, monotonic one, and hence it converges.

Poison

Evaluate this limit.

Next, solve this question:

A typist makes about 1 error in 200 words. What is the probability that he types the 1st 2500 word without error?

Solenoid

A long solenoid with its axis as the z-axis and uniform cross-section perpendicular to the z-axis carries a current I and has a turn density n in vacuum.

Compute the magnetic field it produces everywhere if its cross-section is circular.

Click here when you've done this.

Click here for the solution.

Hourglass

A "water"-hourglass is to be constructed such that the scale is uniform.
That is, the ratio of height difference between the level of water at two time instants to the time difference is a constant.


Your task is to find the mathematical equation the glass follows.. ie, report d log z / d log r accurate upto 3 decimal places at the judge.
Assume the hole to be infinitessimally small.
Also assume negligibe change in air pressure.

Inductor anyone?

A solenoid of self-inductance L, is allowed to reach steady state by connecting to a battery of emf E and internal resistance R.

Now, a soft iron core is quickly inserted in the solenoid such that the coefficient of self inductance changes from L to nL.

Determine the current in the circuit at the time of complete insertion.

Hollow or Solid

Petr has 2 rigid bodies 'A' and 'B': one is a hollow cylinder, the other solid. He allows them to roll (from rest) from the top of a circular track.

'A' displaces by an angle 'Qa' before losing contact with the track. 'B' displaces by 'Qb'. Also Qa>Qb.

Which is which?

Mechanics:

A small ring is located near one end of a rod. Next, the rod is given an angular velocity 'w' along the axis shown.

With what kinetic energy does the ring leave the rod?
Mass of ring='m'. Mass of rod='M'. Length of rod='L'.

The action takes place in a horizontal plane.

Poll:

Please vote here.

Kinematics:

Till yesterday, I thought these problems were difficult:

Find 'x' so that the 2 particles collide.

Moment of Inertia:

Find the moment of inertia of a hollow cone of mass 'M', and base radius 'R'.

PS: I know I'm repeating this problem.

Simple Mechanics:

Here is a practice problem:

2 balls each of mass 'm' are released from the top inside a tube of mass 'M' and radius 'R'. We require that the tube lifts from the ground when the 2 balls make an angle 60* with the horizontal as shown.

Find M/m required to satisfy this requirement.

Light:

A solid iron sphere of radius=1m is just submerged in a water tank. A point source of light is kept vertically above its center at a distance 1m from the water surface.

Find the area of the sphere's shadow. Refractive index of water=1.414.

Here's a Hint

Lessons in Charge Distributions - I

A charge 'q' is placed at a distance 'a' from a very large metal sheet. For a=10μm and q=6nC, report the best integral approximation of the force between the charge and the sheet in Newtons. Assume that area of the metal sheet is of the order ~ 10m^2.


Click here for the solution to the Falling Raindrop question.

Click here for the solution to this question.

Surface Tension:

A liquid rises to 15cm in a thin capillary. The radius of curvature of the liquid's surface (in th capillary) is 1mm.Now, the capillary is cut at a the 7cm mark.

1)What is the new height of the liquid in the capillary?
2)What is the radius of curvature of the liquid surface?

Electrostatic Experience

In a 3-D space, the potential depends only on 'x' and is given by V(x,y,z)=-2*x^n. In other words, the potential is proportional to the nth power of the x coordinate.

In this case, find P(x,y,z), where 'P' stands for the volume charge density.

Click here for the solution

Hydrostatics:

A cuboidal block of iron (6.8g/cc) ans mass=1kg is immersed in water upto 1/2 its height.

A spring with force constant K= 1N/m presses the block from above. Non-deformed length of the spring was 50cm.

Find the normal reaction on the block (at its bottom surface).

Falling Raindrop

A raindrop, whose initial size may be considered negligible, falls from a height h and increases its size by accreting (gathering) all the mist it encounters. If the mist is of uniform density ρ and extends to the ground, report the equation of motion of the falling raindrop.

P.S.: Equation of motion refers to an equation in position, velocity and accelaration.

Click here for the solution.

Ram an RanGen

Ram has a machine which he calls RanGen (Random-Number Generator). RanGen generates random numbers b/w 0 and 1.

Ram is playing a game: He uses RanGen twice and hence gets 2 numbers 'a' and 'b'. Then he shows only 'a' to you and asks the following question: 'Is 'a' larger than 'b'??'.

If you guess the answer correctly, Ram gives you 101 Rs. Otherwise he takes 101 Rs from you. Ram is willing to play this game for ever, everytime with a new 'a' and 'b'.

Develop a scheme for guessing which will 'probably' make you rich in the long run.

Note that Ram is very kind, and will let you borrow RanGen for some time if you want.

Easy? Yes it is!

If you're good, this one should not take more than 15-20 seconds even if you don't remember any formulae.
Even that's too much.

Find the flux, of a charge q, through an infinite sheet placed at a perpendicular distance d from it.

Applying JEE knowledge:

Solutions on the way.

As promised, we're trying to provide solutions for most of the blog questions. Here's the solution to a unique SHM problem.

Only one person was able to solve this correctly, so here's a chance to learn! Its a must try for 11th class students.

Hello World

I will praise myself and say that this is a wonderful blog. We have very good questions for JEE physics here, and few (if not none) books could match this collection.http://www.blogger.com/img/blank.gif

I have been very irresponsible in the past 6 months: I have not written the solution to any problem. In many questions, I see that readers have specifically asked for solutions, and I have not replied.

In order for the blog to be more useful, I'll be posting solutions regularly now, while the other 2 authors will post new questions.
I will also post questions if I find good ones.

We will host the solutions on www.shivankergoel.blogspot.com

Lyttleton-Bondi Model for the Expansion of the Universe

In 1959, Lyttleton and Bondi suggested that the expansion of the Universe could be explained on the basis of Newtonian mechanics if matter carried a net electric charge. Imagine a spherical volume of astronomical size and radius R containing un-ionized atomic hydrogen gas of uniform density η, and assume that the proton charge e(p) = (1 + y).e, where is the modulus of the electron charge.

a) Obtain the value(expression) of y for which the electrostatic repulsion becomes larger than the gravitational attraction and the gas cloud expands.

b) Obtain an expression for the force of repulsion on an atom which is at a distance R from the centre of the spherical volume. Hence show that the radial velocity is proportional to R. Let us label the proportionality constant as H. Assume that the density is maintained constant somehow by the continuous creation of matter in space. Assume also that the value of y is quite larger than the equilibrium value calculated in (a) above and hence ignore gravity.

c) Given that at time t = 0, the volume of the Universe was Vo, obtain an expression for the volume expansion of the Universe.

Note: Experiments do not indicate a difference in the magnitudes of the electron and proton charge. Some theories regarding the nature of the fundamental forces and elementary particles also do not point to a difference. Hence the Lyttleton-Bondi model has been largely discarded by the scientific community.

Fraud

This one might be a common question i guess, but not common enough!

A uniform rod of mass 7 kg and length 120 cm is hinged at one end. A particle of mass 2 kg travelling with a speed 15 m/s collides with the rod at a distance x from the center of mass of the rod such that the reaction force at the hinge is zero.

Find x.

Click here for the solution.

Permutation and Combination:

Consider the set 'S(n)' formed by erasing digits at some (0 or more) positions of a number 'n'.

For example: if n=123, S(123)={123,12,13,23,1,2,3}.
Another one: if n=122, S(122)={122,12,12,22,1,2,2}.

Let 's' denote the sum of all elements of 'S'. Find s%9 for n=12391227.

Note:
1) '%' means remainder. 5%2=1.
Some Identities:
2) (a+b)%c=(a%c+b%c)%c
3) (a*b)%c=((a%c)*(b%c))%c.

Is it possible?

Easy, yet Interesting!

Given are 2 positively charged infinite rods of a finite charge density k.

The rods, both move parallel to their length with a speed 'v' in the same direction. Find 'v', if they are to remain in equilibrium.

Bonus question:- A particle of charge q and instantaneous position r, is moving with a velocity v. Find the magnetic field vector at a point with position vector r1.

Don't Get this Wrong!

Two bobs, tied to each other by an ideal string, lie on a smooth, frictionless horizontal table.

A sudden impulse 'J' is imparted to the bob of mass 4m. Find (x+y), if the tension in the string just after the impulse is [x(J^2)]/[yml].

Use Judge-'Impulse'

This might be tough!

The figure is self explanatory, you just have to find the final velocity the particle attains.

(Pardon my drawing.. The figure shows an inclined plane of inclination 30 degrees, on which a particle is given a velocity 60m/s)

The mathematics might be tougher, so u MAY just report the DE you get.

Friction, Elastic Collisions, and Projectiles

A small spherical ball undergoes an elastic collision with a rough horizontal surface. Before the collision, it is moving at an angle ß to the horizontal.

Find ß as a function of µ, such that the subsequent range is maximised.

Now determine the maximum value of ß for range to be maximum. What if ß is more than that maximum value??

Report your answer at the judge in degrees, rounded off to the nearest integer.(The judge checks only the maximum value of ß.)

I know its easy, so don't bother setting a record

Find the time period of small 'torsional' oscillations. The mass of the solid hemisphere is 'M'.

Click here for the solution.

Best of Luck:

Best of luck to all readers who gave JEE.

Have you seen this:

There are 2 uniformly charged spheres (call them charge clouds):






Sphere 1: volume charge density: P, radius: 1, center: (0,0,0).
Sphere 2: volume charge density: -P, radius: 1.5, center: (1,1,1).

Find the electric field in their region of intersection.
Answers welcome.

Roll

A block of uniform mass M = 2.5 kg is at rest on a table. A disk of mass 2M, radius R = 60 cm and of the same height as the block, which is initially spinning about its axis with angular speed w = 3.5 rad/s, is placed on the table such that it touches the block.

The block-disc system thus starts moving such that they are in contact throughout the motion.

a) What is the initial accelaration of the block-disc system?
b) Determine the instants of time t* and t(tot) when pure rolling starts, and when the block comes to rest, respectively.

Friction between the disk and the block may be ignored, and coefficient of friction (static as well as kinetic) at all other surfaces of contact is µ = 0.3. Take acc. due to gravity to be g = 10 m/s^2.

You may check your answer at the judge in the format "5,4,6" if your answer for part a is 5 m/s^2, t*=4 deciseconds and t(tot)=6 deciseconds.

Awesome

An electrostatic field line leaves at an angle α from a point charge +Q, and connects with another point charge -q at an angle ß.(See Fig.)

1.)  Suppose Q = 12 µC; q = 6 µC; and, α=60*. Determine ß.

2.) Determine the general relation between Q, q, α, and ß.

Click here for the solution.

William Pickering:

The dual-wedge shown in the figure is MASSLESS. All surfaces are smooth.
Find the min ratio m1/m2 so that the ball m2 'starts climbing up the wedge'..ie. it starts moving up the incline in the direction shown.

Problem Statement Courtesy: Ashish Gaurav, the second-best physics guy I've ever seen.

Mirrored

A particle moving with initial velocity u = ( 3i + 5j ) units collides with a smooth plane wall placed at some orientation to the particle's trajectory such that the resulting velocity of the particle is v = ( -2i - j ) units.

Determine the orientation of the reflecting plane.

Kinematics: Polar Coordinates?

I did this question in the last week of my JEE journey. This question is all mathematics-vectors-coordiantes-derivatives, etc, but one needs firm foundations on the significance of angular velocity, and 'radial' velocity

A particle, initially at a distance d from the origin on the X axis takes off on a peculiar trajectory, driven by an external agent dealing with which is none of our bussiness. The trajectory is such that at any position of the particle (r, Q) in polar coodinates, its velocity is [(-VocosQ icap) + (Vo(1-sinQ) jcap)], where Vo is a constant

We are to find its angular velocity about the origin, its radial velocity about the origin, and its distance from origin as it strikes the Y axis, in terms of the bold data

Free question-> a particle at (3,4) is moving with velocity (1,2,3), find x+y+z, if its angular velocity about origin is (x,y,z)

-Sambhav

Packets

This one uses a great technique I've codenamed "packets". (Can you figure it out?)

Two positrons and two protons are kept on four corners of a square of side a = 1 fermi.

The mass of the proton is much larger than the mass of positron. Then determine the kinetic energies of one of the positrons and one of the protons respectively after a very long time.

Note: Its more of a mechanics question than electrostats.


P.S.: Report your answer at the judge in micro-ergs, correct upto 2 decimal places. eg - "0.58,6.50" if ur answer is 0.58 microerg for a positron and 6.50 microerg for a proton.

P.P.S.: You may take K = 9x10^9, e = 1.6x10^(-19) SI Units.

-Shivanker

Click here for the solution.

Polytropic Process

Any thermodynamic process, represented by a Gas Equation in terms of product of powers of parameters {P, V, T} equated to a constant is called a polytropic process.

For example-

(P^3)(V^5)=constant

(P^-1)(V^-3)(T^6)=constant

(P)(V^gamma)=constant...(yes, even the adiabatic one)

P=constant

V=constant, and so on

now, i take a general process (P^a)(T^b)=constant, or equivalently, (P)(T^b/a)=constant, calculate

1)molar heat capacity in terms of Cv, R, a, and b

2)coefficient of volume expansion in terms of instantaneous temperature T, a, and b

3)bulk modulus in terms of instantaneous pressure P, a, and b

and

4)learn if possible..(the resemblance in the 3 is uncanny, and always saves up maths and calculations worth 2 mins during the exam, a double treat if this equation comes in a paragraph!)

now, if someone comes up with a process (P^l)(V^m)(T^n)=constant, use the ideal gas equation to convert it in the form (P^a)(T^b)=constant, and then use the formulae we learnt :-)

-Sambhav

Dog vs. Cat

Now this is an awesome question.

A cat is running in a straight line at a constant speed u.

Now a dog sees the cat when their line of separation(=d) is perpendicular to the line of motion of the cat and starts running after it at a constant speed v, such that it's always headed towards the cat.

1.) Find the total time taken by the dog to catch the cat if v = 3 m/s; u = 2 m/s; and, d = 10 m.

2.) Find the final separation between the 2 animals, if d = 10 m; and, u = v = 3 m/s.

P.S.: To check your answer at the judge, use this syntax without the inverted commas(ie, if your answer is 6.32 and 5): "6.32,5.00"

Click here, for the solution.

Practice:

Recently, Sambhav made a post about a nice method to solve SHM questions. I remember getting frustrated over such problems: the solution always seemed a bit too lengthy.

Here's your chance to get a firm grasp over the method. Try finding out the period of oscillation.
I request Sambhav to post the solution after 2 days or so.

(I hope that this one can be solved using the method).
-Anuj

Probability :- Survival?

This SHOULD be fun!

one of those questions which tell you that maths IS beautiful :-)

this has two solutions, try to get both of them, you'll see what i mean!

Let 'm' the probability that a man aged 'x' years will die within a year. Let A1, A2, A3, A4......An be 'n' men aged 'x' years. Find the probability that out of these 'n' men, A1 not only dies within a year, but also is the first one to die.

-Sambhav

SHM :- Approximations

Only looks difficult: Find X, if the time period of oscillations when the rod is 'slightly' disturbed towards one end is 2pi/X.

You may use the judge

-Sambhav

A Rope

This one is another easily confusible problem (maybe not for some of you).
A long flexible inextensible rope of uniform linear mass density 5 kg/m is being pulled on a rough floor, with a horizontal force F, in such a way that its lower part is at rest and upper part moves with a constant speed of 2 m/s.

What should be the magnitude of this force F ??

-Shivanker

springs are in series!

A single string went through all the pulleys, so all springs had the same tension in them, that is, k1x1=k2x2=k3x3=k4x4=2T

and recall, when springs are in series, the same force acts through them!

So, the whole problem reduces to this

or if one wants to spend a lil more time but wants to go through basics, he should form these equations

1)T=ma...(neglecting mg for calculation of time period)

2)equilibria of massless pulleys; k1x1=k2x2=k3x3=k4x4=2T, k1=k2=k3=k4, means x1=x2=x3=x4

3)length constraint, 2(x1+x2+x3+x4)+x=0, where x is displacement of block

Using these equations, find a relation of the form Kx=-ma, determine K, and all is done

yes, X=4

Constraints? NOT ALWAYS!

For m=10kg, and k1=k2=k3=k4=10N/m, find X, if the time period of oscillations of the block is 2piX seconds.

Try doing the question not as a challenging physics problem(it isn't challenging either :P), but as a jee question that needs to be done in less than 2 mins.
Needless to say, the pulleys, string and the springs are ideal.


-Sambhav

Resonance

This is a question I usually ended up confused at.

A uniform tube 60 cm long, stands vertically with lower end dipping into water. When its length above water is 14.8 cm and successively again when it is 48 cm, the tube resonates to a vibrating tuning fork of frequency 512Hz. Determine the lowest frequency to which this tube can resonate when it is taken out of water approximated to the nearest integer.

(You can check your answer at the judge.)

-Shivanker

"The old order changeth yielding place to new".

Masters Shivanker Goel and Sambhav Gupta have kindly agreed to write for the blog.
Their passion for physics is evident in the number and quality of questions they have solved here (and probably in their JEE performances too).

Expect cool questions from them soon!

Collisions:

Consider a particle surrounded by a circular wall. (Top View is shown)

A particle is projected at an angle 37* with the radial direction. It returns to the point of projection after 2 collisions. Find the coeff. of restitution for the collisions between the particle and the wall.

Triple Filter Test:

Before you say something serious, try the Triple Filter Test:

In ancient Greece, Socrates was reputed to hold knowledge in high esteem. One day an acquaintance met the great philosopher and said, "Socrates, do you know what I just heard about your friend?"
"Hold on a minute," Socrates replied. "Before telling me anything I'd like
you to pass a little test. It's called the Triple Filter Test."
"Triple filter?"
"That's right," Socrates continued. "Before you talk to me about my friend, it might be a good idea to take a moment and filter what you're going to say. The first filter is Truth. Have you made absolutely sure that what you are about to tell me is true?"
"No," the man said, "actually I just heard about it and..."
"All right," said Socrates. "So you don't really know if it's true or not. Now let's try the second filter, the filter of Goodness. Is what you are about to tell me about my friend something good?"
"No, on the contrary..."
"So," Socrates continued, "you want to tell me something bad about him, but you're not certain it's true. You may still pass the test though, because there's one filter left: the filter of Usefulness. Is what you want to tell me about my friend going to be useful to me?"
"No, not really."
"Well," concluded Socrates, "if what you want to tell me is neither true nor good nor even useful, why tell it to me at all?"

This is why Socrates was a great philosopher and held in such high esteem. It also explains why he never found out his best friend was sleeping with his wife.

Light

A ray enters a glass sphere which has one of its sides silvered. It emerges from the sphere after 1 reflection from the silvered surface. In this process, the ray suffers some deviation. The deviation angle is found to be minimum at i=60 degrees.

What is the refractive index of the glass correct upto 3 places of decimal?
Note: You can check your answer using the judge on the left

Basics:

This question belongs to a special type of physics questions. They involve a 50-50 blend of maths and physics.

A small block (1kg) slides down the smooth circular surface of a wedge (5kg).
What should be the minimum coeff. of friction b/w the wedge and ground such that the wedge does not slip during the motion of the block??

Series

There are 100 points in 2D space :(x1,y1), (x2,y2), .... , (x100,y100).

Write a formula for the sum of squares of distances between all pair of points.

The number of terms in the formula should be as small as possible.

JEE 2011:

Here's one concept that was highlighted in the paper. Literally, people (including me) fought over the validity of the problem. (Q 33, Paper 1).

Lets not get into the problem. Instead, lets do something simpler.

The vertical black rod is mass-less. The blue rod has mass 'm'. The vertical rod is rotated at an angular velocity 'w'. Calculate the kinetic energy of the system for two cases:

1)Blue rod is free to rotate about 'A'. ('A' is then called a 'hinge' joint).
2)Blue rod is not allowed to rotate about point 'A'.

P.S. Hope the blog readers did well!

Good Physics Questions:

Best of Luck to all giving JEE tomorrow! Lets hope that they give some good physics questions!

Polygon Area:

A simple polygon on 'n' vertices has sides AB,BC,CD,...,NA the number of sides being 'n'. The polygon is 'simple', meaning that it can be convex or concave but not self-intersecting. A,B,C,...,N are the n-vertices.

How can you find the area of the polygon, provided that you can compute only 'n-2' cross-products of vectors?

For some polygons, google 'concave polygons'.

Area Bounded:

Consider the isotherms drawn in the F(E) vs E graph.

Find the area bounded by the isotherm at T=0K and the E-axis.

Notes:
1)'E' means energy.
2)'T' is temperature, 'k' is your Boltzmann constant.
3)F(E) is the equation for the Fermi-Dirac statistics.
4)The above 3 are insignificant for the problem.

The horizontal bottom of a wide vessel with an ideal fluid has a round orifice of radius R1 over which a round closed cylinder is mounted, whose radius R2>R1. The clearance between the cylinder and the bottom of the vessel is very small, the fluid density is p. Find the static pressure of the fluid in the clearance as a function of the distance r from the axis of the orifice (and the cylinder), if the height of the fluid is equal to h.

Error Analysis contd..

Rules:

1)'D' is the distance b/w the virtual sources and the eyepiece, measured with a scale of least count 1mm.
Procedure: The virtual sources are at '0' on the optical bench (no zero error). The coordinate of the eyepiece is taken.

2)The fringe width and 'd1' and 'd2' (displacement method values) are measured with a micrometer screw of LC .01mm.
Procedure:: The cross-hair is set at a good- dark fringe (or at the image of a virtual source). Then it is moved to the 25th fringe (or to the image of the other virtual source). The difference between the 2 readings is taken.

Find the wavelength of the light used. What light is this?