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'.