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