Frustum

A unit charge is placed inside a hollow frustum as shown. Let the flux through the curved surface of the frustum be 'F'.

Report the value of F*epsilon0 correct upto 2 places of decimal. Note down the number of WAs before you get accepted.

A New Year Gift:

Introducing a new feature: Check your answer!
To your left, you should see something asking for your answer. This is automated, so it needs a syntax:

sin(A) is good. Sin(a)/SIN(A)/Sin(A) etc are not..
log(A), ln(A), exp(2) are all good. Don't write e^x.
'g' is g. Its not 9.8, unless specified.
In the question field, choose the question title.

For example: the answer to the last question was "both" (quotes for clarity). The title was "A Frozen Lake". Try it. If it does not work (which is probable as I wrote the code), leave a comment.

A Frozen Lake:

A common concern for anyone who visits the Dal Lake and wants some fun is : "Is the ice safe enough?". The expert's answer is, well, the ice is never safe enough.

But here we'll adopt a more scientific approach. It is well known that
snowmobiles and ATV’s need at least 5 inches, and cars and light trucks need at least 12 inches of good clear ice.

Here are some stats of the Dal Lake:
1)Total depth (water+ice) in winters: 20ft.
2)Surface temperature: -16*C.
3)Bottom temperature: +4*C.

Assume the thermal conductivity of ice: 2.18 units, and that of water: .58.

So, is the ice safe for snowmobiling? for driving? for both? for none?

Angle with the horizontal:

A massless rod has a 'mass' glued to it. The position of the mass is 1m from the left end, and 3m form the right end.

This rod-mass combo is placed on the frictionless inclined-planes shown. At equilibrium, it makes an angle 'B' with the horizontal. What is this angle?

PS: What is the number of cuboids in a 'Rubik's Revenge'? (Ignore the internal structure of the toy.)

To see a Rubik's Revenge: See This

CM:

This is a solar panel (believe it). Your company proposes to set it up in some remote desert. You have a plan:

You want to have the panel balanced on the tip of a thin vertical tower (!!!). Give the equation of the tower wrt. the given coordinate axes.

(I kid you not. I'm not making this stuff up!)

This is One-of-the-Best:

A point source of light radiates monochromatic light with an intensity 1 Cd.

Now, take a convex lens (f=10cm) and place this source at its focus. (You have one focus left.)

Simple, whats the intensity at the other focus?

Assume lens is too small..:P

Textbook:

This is a fairly standard problem, which most of you might have seen.

A flexible cord is placed on a quarter-circular surface (radius=1m). With what speed does the bottom of the chain hit the ground?

A Flyball-Governer:

The diagram shown is that of a "FLYBALL GOVERNOR". It consists of 2 bearings 'A' and 'B'. 'A' is fixed while 'B' can slide up and down the rod.

The bearings are attached to 2 massive balls (assume 1000kg balls). The balls are spinning with a speed 10m/s.

The speed of the balls can be 'governed' by adjusting the distance between the bearings.
The distance is reduced (slowly and smoothly) form 1m to 1/2m.

What is the new speed of the balls?


NOTE: ignore the '3m' shown in diagram.

Open your third eye:

Find out unequal positive integers a,b,c,d,e,f,g,h such that
(a+b+c+d+e+f+g+h)^2=
a^3+b^3+c^3+d^3+e^3+f^3+g^3+h^3

There is man named 'Mabu' who switches on-off the lights along a corridor at our
university. Every bulb has its own 'toggle' switch that changes the state of the light. If the light is off, pressing the switch turns it on. Pressing it again will turn it off.
Initially each bulb is off.
He does a peculiar thing. If there are 'n' bulbs in the corridor, he walks along the corridor back and forth 'n' times. On the 'i'th walk, he toggles only the switches whose position is divisible by 'i'.
He does not press any switch when coming back to his initial position.
The 'i'th walk is defined as going down the corridor (doing his peculiar thing) and coming back again.
Assume n=44,100.
Determine the final state of the last (44,100th) bulb. Is it on or off?

The 1st, 2nd and 3rd quadrants of the Cartesian Coordinate plane are given a charge density 1C/m^2.

What work has to be done in taking a unit positive charge from z=1 to z=3?

I Love This One:

A hoop (mass 'm') is placed on a smooth horizontal surface as shown. A particle (mass '2m') is initially at the position shown, then it is given a velocity 3m/s, tangential to the hoop.

Radius R=(1/pi)m.

The motion that follows is quite complex, so is your analysis.

Find the time in which the particle completes 1 complete revolution inside the hoop.

Clarification:

Here's a little argument as to why the charge density increases as the radius decreases:

Consider two conducting spheres connected by a wire.
Let us give the 'system' a charge Q. This charge will distribute itself between the two spheres.

Finally, let the charges on the two spheres be q1 and q2. Can you prove that:

q1/(4*pi*r1^2) is greater than q2/(4*pi*r2^2)? This implies that the charge density is higher on the smaller sphere.

I am trying to remember some 'good' maths problems, but I am having a tough time.
Till then, try this:

Charge 'Q' is given to a hollow conducting cone. Find the electric field at the center of the base of the cone.

Assume any parameters that you need.

As promised, here is another problem from our ALM110 major exam:

The diagram is self-descriptive. The 'hoop' rolls without slipping and its center moves to the right with a velocity 'V'. The radius of the hoop is 'R'.

Find the angular velocity of the rod.

Note: I have added a diagram describing the connection between the 'rod' and the 'hoop'.

Probability:

Consider a collection of families, each of which has exactly two children. Each of the four possible combinations of boys and girls, bg, gb, bb, gg, occurs with the same frequency. A family is chosen uniformly at random, and we are told that it contains at least one boy. What is the (conditional) probability that the other child is a boy?

AML110 major:

You could have probably solved at least two problems from our AML110 major exam.

Here's the first:
A rod is constrained to move b/w two fixed cylinders. The cylinders however are free to rotate about their natural axes.

Find the acceleration of the rod.

Radius of the larger cylinder = .6m.
Radius of the smaller cylinder= .3m.

Boating:

A 100kg man(standing on shore) wishes to start his journey in a 200kg row-boat (parked in the water near the shore) with good initial speed. What he does is this:
1)He heaves the boat forward (with great effort) providing it with 2 units speed.

2)Next, he sprints on the beach sand, and jumps into the boat (his running direction coincides with the boats motion), with a velocity 5 units.

What is the final speed of the boat??

You'll encounter similar questions in your JEE.

The rod (weight=W) is hinged at A. It is released from rest from the position shown in black.

1)What is the reaction from the hinge at this instant??

2)What is the reaction when the rod has rotated by 45 degrees?

CSL105 quiz2:

A bag contains either a 100 Rs note or a 500 Rs note, with equal probability.
I add a 100 Rs note to the bag. (I'm rich).

'A' draws a note from the bag and it is Rs 100. 'A' keeps it.

'B' draws a note now. What is the probability that 'B' draws a 100 Rs note?

Answer=2/3.

The upper rod is hinged to the roof; the lower rod is hinged to the upper rod.
Mass of both rods=9kg; length=1m each.

Now, a force 45N is applied to the upper rod as shown. At this instant, find the angular acceleration of the lower rod.

Amendments:

Here's something to make things a wee bit clearer:

A rod of mass 'M' and length 'L' is pulled by a pretty large force 'F' in the manner shown. (In the lower diagram).

(All this happens on a horizontal table...The rod is hinged to the table's vertical edges..).

Find the time period of small oscillations of the rod.

OOPS!!!!!!!!!

Now what's this???

All the wires shown in the figure (red color) have resistance 'R'. The only places where the wires are connected are the vertices of the large hexagon.

Find the resistance between points
1)'A' and 'B'.
2)'A' and 'C'.

Friction:

What is the maximum weight of the block B which can be supported in the manner shown? (The pulley is smooth).


Note that the rod is nailed to the disc in the "black" region. The rod is also massless.

The coeff. of friction at the contacts b/w disc and surfaces is .3.

A rectangular block of mass 1 kg is trapped below a wedge (mass=10 kg). The dimensions of this block are negligible compared to those specified in the diagram.

One plans to take the block out by applying a pulling force to the left.

Answer the following questions:

1)What minimum force must be applied to accomplish this?


2)If this bare minimum force is applied:

a)What is the initial acceleration?

b)If the same force is applied to the right (pushing), what is the maximum displacement of the block?

c)What is the accn. when the block starts to come out of the bottom of the wedge?

d)If one continues to apply the same force for a very long time, what is
the acceleration at t=inf.


Assume the coeff. of friction at all contacts= .5.

Count:

Suppose that a weapons inspector must inspect each of 5 different sites twice, visiting one site per day. The inspector is free to select the order in which to visit these sites, but he cannot visit the site 'X' on 2 consecutive days.

Find the total number of ways in which the inspector can schedule his visits.

Here is a fixed circular track with a ball in it. The ball is projected with a speed V0 from the bottom of the track. This speed is more than that required to complete a circle.

Let normal reaction on the ball at the top of the track = N1.
At the bottom Normal reaction = N2.

Find N1-N2. Assume mass of ball = m.

Tark:

Shown is an iron rod supported by two vertical threads.

Mass of rod=M.
Separation b/w threads=L
Angle made by rod with the horizontal=Q

The tensions in the two threads are T1 and T2. Find T1/T2.

Because I can't think of anything better:

A hollow cylinder (radius=.5m) rolls without slipping down an incline. The length of the incline is 1m and the friction force acting on the cylinder is 200N.

Find the rotational work done by the friction force on the cylinder during its motion on the plane.

AML110 quiz:

The beam (in black) supports a parabolic laminar sheet . The weight of the sheet is 200N.
Assuming the beam to be massless, find the normal reactions from the two wedges.

The Kings Of S

I'll line up some tough ones now. These are definitely NOT for the weak-of-heart.

'n' points in 3D space are connected to each other by capacitors. (Every point is connected to every other point.)

Two points are selected at random. What's the capacitance b/w these two points?

Long time, no question...

Easy one here:

The ball is released from rest in the semicircular groove as shown...The wedge with the groove (radius 'R') has mass 'M', the ball 'm'...

The ball slides down the groove (gaining speed) and then reaches a maximum height on the other side of the groove's center (losing speed here)...At this posn., find the velocity of the ball.

Congrats!!

Many people cleared the JEE today..congrats to em all! Further, the blog now boasts of 60 followers..so congrats to me.

Now, this question HAS a history with me..
Particle 'A' is located at (1,2,3), and particle 'B' at (-1,-2,6). A has a constant velocity V1(vector), B has V2(vector).

Find the unit vector in direction of (V2-V1), if A and B are supposed to collide somewhere in the future.

This is difficult:

A marble rolls down (from rest) from a cliff of height 5m, and then encounters a vertical drop of 1m.
Find the range of the marble on the horizontal planes...

For those of you who don't know already, marbles are solid spheres, and are quite small..

Properties that Matter:

No reples to the Sand's question??? Wassup?

Now, this is something we all can do:
Shown is a spherical shell of some conducting material, like say, Steel. The inner surface is maintained at 1000K and the outer at 2000 K.

The radius of the inner and outer spherical surfaces are 1m and 2m. Find the temperature at the spherical surface of radius 1.5m.

Sands :

Here is a method to determine the coeff. of friction b/w sand particles:

A circle of unit radius is drawn (with chalk) on the ground and sand is sprinkled from a hopper into it..the maximum height of the cone achievable thus is H.

Find the value of dU/dH (U : the coeff. of friction), in terms of H.

Solution:

Not many were able to solve the length of the cycloid...
Don't worry, i never solved this one myself!!! Had an hour of frustration before looking at the solution...

Here's my version of the solution:
If you are into center of rotation, you can easily write the velocity as a function of time:
Here, the length CP is 2RCos(90-theta), which is 2RSinQ..
Q is infact wt...so V=[2RSin(wt)]*w...as the velocity is equal to the distance from center of rotation*angular velocity..

All that remains is to integrate V(t) from 0 to 2pi/w...

No partial marking:

This is the sort of problem that you will see...someday.

The solid cylinder shown in the figure is of mass 5kgs and radius 25cm. It is kept on two rail-like planks and a force 15N is applied as shown..

What is the force of friction on the cylinder from one plank??

Cycloid:

The integration involved in the calculation of length of the involute was (after some thought), intuitive...there was little choice!!

Here's something better: the circular object rolls without slipping on a horizontal surface. The motion of the lowermost point 'P' is monitored.
The path traced by this point is called a 'cycloid'...again: find the length of the cycloid, assuming the radius of the circular body 'R'...

Maybe this will do better:

Free Counters

The last question drew little response, so this is something better (with a counter, which sucks):

This is a fixed vertical cylinder (top view), and it is connected by a thread (of length equal to the circumference of cylinder) to a ball.
Now, the ball is wound around the cylinder, keeping the string taut at all times.

The path traced by the ball is called an involute...simply speaking: find the length of the involute.

One from the good old days:

I solved this one a few weeks before the JEE...and it made quite an impact...took me approx. 20 mins to even get started...

Here'e the deal:
We have a particle and a ring, both smooth, on a horizontal plane. As shown, they are initially in contact and at rest, when the particle is given a velocity V (as shown).

Find the time in which the particle completes one full revolution inside the ring.

IIT D CYL110 quiz 1

Consider this: this problem appeared in our CYL110 (a 2nd semester chemistry course) quiz...half the CS department couldn't get it right!

This gas undergoes a reversible process s depicted in the P-V curve...the center of the circle shown is (10,10) and its radius is 5;

You know whats coming...find the maximum temperature of the gas!

Fluxes and Fields

A disc (radius R, surface charge Sigma), and a point charge Q are separated by distance x...

Do:
1)Find field at the location of the charge due to the disc.
2)Flux due to the charge's field thru the disc.
3)Learn them both...you already know (1)!

Let's Begin:

This is a fairly simple one:

Given is a uniform line charge of shape as shown. Find the value/(s) of the angle A such that the electrostatic field at the center (of the loop/arc shown ) is zero.

I'm Back...

With more than two months free ahead and virtually nothing to do, i guess i'll make an attempt to revive this blog...If you ppl have done electrostats, then leave a mesg. I am out of mechanics problems, as is evident from this problem, which i copied directly from the 100P forum..

This problem teaches a few things, and though the mechanics involved is simple, the thought process is cool:

A solid sphere rolling without slipping on a rough horizontal surface with a linear speed 21m/s collides elastically with a fixed, smooth vertical wall. Find the speed of the solid sphere after it has again started pure rolling (in the backward direction) .The radius of sphere being 1m.

This ain't textbook:

All the rods in figure are hinged smoothly as shown. The vertical red line is infact a spring of force const 'k'...

In terms of the angle A, k, m find the period of small oscillations (along x) of the blocks.

Textbook:

This is a standard textbook question:
The spring-block shown falls from a height of 1m. Assuming the spring to be sufficiently long answer these two questions:

1)What is the maxm. speed of the block?
2)What is the amplitude of oscillation?

Anyone who answers both correctly???

SHM..finally.

The block (mass 5kg) is released from rest when all springs are relaxed...

It undergoes SHM...Find the amplitude of oscillation of the point P.

Tis simple:

The grey blocks are of mass 1kg and the red one 2kg... the force F is 35N...all surfaces have coeff. of friction .5.

Angle A=37*.

Obviously, find the accn. of the grey block.

WARNING : THIS IS TOUGH:

If someone does'nt have a liking towards really tough ones, plz. take a walkover...

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.

Seems trivial, but ain't:

The answer to this problem should act as an eye opener to the simplicity of many things;

Assume: m1 moves downwards. Find the accn. of m1.
Bet you had to frame some (2) equations...i don't...
Try to observe something in the result. Report anything unusual;

Running out...

Try this:

The rigid body is so selected from a collection of spheres, rings, discs etc. such that when released from rest in the position shown, it leaves contact with the horizontal plane at exactly 60*...as shown.

This would assist in a smooth 'transition' from the horizontal to the inclined surface;

So, whats the nature of the rigid body???

The simpler, the merrier.

I' scrapping this question...you would never solve it as it seemingly can't be solved...i tried some graph plotting software but could make nothing...some people more skilled in trigo. would have done it maybe...

And, as you can see, i'm really out of cool problems. I'll try to make some oer the weekend.
See you on monday.

The length of both the rods in the figure is 1m. The rods are smoothly hinged.
Find the velocity of the plunger in the cylinder.

Who stole my L???

A cubical block is given an initial velocity V on a rough horizontal floor...After some time, it is found to stop.
The angular momentum of the block about any point on the ground is non-zero initially but zero finally. Which force is single-handedly responsible for this loss:

A)Normal reaction.
B)Friction
C)Weight of the Block
D)Coriolis force.

Just for kicks:

A ball of mass 4kg is projected and the range is found to be 10 meters..
The same ball is again projected with the same velocity and angle, except that this time, at the highest point of its trajectory, it explodes into 2 fragments of mass 1kg and 3kg...the instantaneous velocities of both masses just after the explosion is horizontal..

Finally, the mass 1kg is found at a distance of 3m from the point of projection...(assume that the fragments stick to the ground on contact)
Specify the location of the 3kg fragment...

Plz. note:

In the first diagram, note that though v1CosA=v2, same constraint does not hold for accn.

In the second diagram, v1(1-CosA)+v2=0 but not a1(1-CosA)+a2=0...




I wanted you to observe that you can relate the velocities by taking the tensions..



Try it: in each question, observe the tension (in the dirn. of motion) and the coefficients in the velocity constraints...

Blue...

Somebody made a giant spherical cavity in the earth such that the earth's center and a point in the surface are diametrically opposite.

The someone drops a small ball from a small opening at the surface into the cavity. In how many minutes does the ball reach the center of earth???

Let Tension lead the way..

Relate the accns. of the masses in the diagrams.

In the 1st diagram, the thread goes through a very small ring, free to slide on a fixed beam..You have to relate the accn. when the angle (specified in dig.) is A.