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