Its Yours:

The center of rotation method:
Apply when



1)ONLY 1 rigid body, with a center of rotation.


2)Take torques about the center of rotation, assuming all bodies to be at rest in equilibrium.


3)The moment of inertia is calculated by assuming all masses to be present as point masses where the thread is wound/ plank is kept.

With each figure, i have attached expression giving the angular accn. of the rigid body..Note that i wrote them all without lifting pen or paper in a matter of seconds..

Give it some practice..try to first actually derive these expressions..only then you'll have a command over it.



Plz. remember that the moment of inertia of a rigid body about a COR is MK^2+Md^2, where d s the distance of the COR from the COM.

Four particles are placed at the corners of a square of side 1m. At t=0, they start moving towards each other (each one moves towards its clockwise neighbor) with a velocity 2m/s.

After traveling half the distance to the center (where they would have collided), each of them doubles its speed.

After what time do the actually collide??

Lets turn on the heat!!

I have masses 2kg, 1kg and 1kg connected as shown...assume all threads to be really very long.

The mass 2kg is released from rest at t=0.
Whats its accn. at t=0??
Whats its velocity after a very very long time?

And one question..which of the following topics are you done with (Lets go for a majority): electrostatics, light (waves waala), SHM...

Some Irodov:

As i'm feeling a bit out of good questions, let's try out these good problems from Irodov:

A car moves on the road in the form of the curve y=Sin(x)...x and y axes constituting the horizontal plane.
It moves with a constant speed. The coeff. of friction b/w the road and car is 3/4.

Whats the maximum speed with which the car can move without skidding?? Neglect all rotational effects.

BOOM!

I have three balls of masses 3m,4m and 2m of negligible dimensions; I want to generate heat using them and a collision;

These balls are kept on the rim of a fixed smooth hemispherical bowl and dropped at the same time; there is a collision at the bottom and i want maximum heat from it;

Specify the orientation of the balls;
Assume that they stick together after collision;

I'm not so happy?

May i ask why no one has given the center of rotn. questions. a shot..well except piyush.

If you are unable to solve individual problems, leave a msg..i'll post one or two demo solns..

Continued...

For those who have not been able to see it till now..and to those who saw it but could not appreciate it:

Solve for the angular accn. of the rigid body..report any similarities in all answers..

Something conceptual:

A thin ring of radius 2m is spun with an angular velocity of 10rad/sec and kept to roll (like a tyre) on a rough floor with coeff. of friction .5;

Its subsequent motion is analysed for the next 1 minute;

Find the time duration for which:
1) the friction acting is kinetic in nature;
2) the friction acting is static in nature;

Insect:

This is a really cute problem:

This poor insect was forcefully kept on a mass less triangular structure, free to rotate in the vertical plane. The only significant dimension of the structure is specified in the diagram.

To save itself from falling, Mr Insect decides to use a simple strategy..he maintains a peculiar motion which does not allow the triangle to rotate!!

Kya aap ek insect se jyaada intelligent hain?? Can you figure out the details of its motion??

Finally, the center of rotation theorem:

In each case, find the angular accn. of the rigid body assuming that it rolls without slipping at pt. of contact with plane or thread...

and observe....


Post your answers for M=2;m=1;R=2;r=1 AND K=1/2;


the answers are not important,; i want an observation...a weapon with which you can solve all questions with a center of rotation simply by lookin at them...


Expect 10 more such problems tomorrow...

Solution:

Minors:

At IIT we have minors every other day, it seems. So guys, can't make another post till maybe next Friday...(2 days for a 400 liner code ain't enough dudes..)

So, try this one out...and wish me luck!!!

The tao of rigid body constraints:

Whenever two bodies are in contact, their relative accn is parallel to the palne of contact.

Calculate the accn. of the plank of mass 'm' in the 1st diagram. Angle of the wedge is A.

In the second diagram, find the accn of the rod kept on top of the rings. Assume the rings roll without slipping.

Description:
i have shown the front and top views of the system; consisting of a short and thin rod kept on top of two rings whose planes are parallel to each other, but the centers are displaced.

The rod moves vertically down and there is always symmetry wrt. the two rings. Find the accn of the rod when the angle that the dotted lines make with the hoz. are 'A' each.

Looks difficult, but ain't:

The system shown consists of a disc and a ring, of equal mass 1kg, and equal radius 1m. Their tops are connected by a smooth mass less thread which winds over them.(as usual)

A time varying force F=kt is applied to the center of the disc. What is the force of friction acting on the ring (in terms of k and assuming that it is rolling without slipping) and in which dirn. does it act???

Let us finish Zone Transition:

For all round rigid bodies, define the zone transition line as follows:

Suppose a Force 'F' is applied at a height 'h' abv. the ground;






1)if h is less than H then friction acts opposite to the dirn. of application of force.
2)For h is greater than H, the friction acts in the same direction as the applied force.

For a rigid body with parameters M,K,R, specify the location of the Z.T.L.

Now See This:

Calculate the thermal resistance of the hollow frustum for the terminals shown, ie the inner and outer surfaces of the hollow frustum...

Assume conductivity to be 1.

Textbook:

Here is a combination of two textbook question:

In the first diagram, the wedge is smooth. The block, when imparted a velocity 'v', achieves a maximum height of 8m...

In the second question, the plank is rough. Given the same velocity, the block travels a distance of 6m on the plank.

Find the coeff. of friction of the plank.
Is such a plank feasable???

An easy one:

This is a standard textbook question:

Find the resistance of the hollow frustum b/w the points P and Q..
Assume:
Resistivity=1;
Outer and inner radius of the left end = 2,1;
Of the right end=4,2;

I need answers to the Electric Field Calculation question, here's a hint: use Symmetry.

Dan Brown anybody:
The illumainati believed in duality of nature, and the were very sure that the ellipse, and not the circle was the best representative of symmetry of nature.

This may be why:

Consider the given table: a ball is fired from one of the foci with a velocity of 5m/s, at an angle 30* with the major axis of the elliptical table.

The question:
When does it again return to the same point as which it was fired from, for the very first time??? (does it ever).

Assume the major axis of the ellipse to be 10 meters, and its eccentricity is .8.
Assume all collisions to be completely elastic.

Find the field: (I need answers)

Find the magnitude of electric field at the center of the hollow sphere of which the shown watermelon-like slice is a part.

Assume the surface charge density of the curved surface to be unity.

Trendz in normal reaction

The resultant normal of all infinitesimal reactions MUST pass thru part B to neutralize the clockwise torque produced by friction.

The trend of normal reaction is shown..this trend may be linearly increasing, but i cannot already say.. clearly the reaction MUST be more on part B, else the resultant WILL NOT pass thru B.

Alpha Scattering:

In the schematic diagram of the alpha scattering experiment, what is the final or terminal velocity of the alpha particle:

1)2*10^6m/s
2)10^6m/s
3)Finally the alpha particle comes to rest.
4)None of these

This will send you spinning for a while...

A meter rod, free to spin abt a stationary vertical axis passing through one of its ends is kept on a rough table of coeff. of friction .5, and spun with an initial angular velocity of 1 rad/sec.

Whats the time taken for it to stop rotating???




Now the rod is replaced by the following object:
A sector (pizza-share like) of a circle of radius 1m and of very small angular width..(say .5 degrees, which means you neglect it).
Again find the time....assume axis passes thru the would-be center of the circle..

Finally do the same for a meter rod whose linear mass density varies as the nth power of the distance from the axis.

This too!

What is the velocity of the ball when it reaches th bottom of the bowl?

Assume its collision with the bowl to be completely inelastic. The accn. due to gravity is g.

You've seen this bfore!

If L/V=t, then after what time does the wedge start toppling off the edge of the cliff??

The hoz. plane is smooth.

The Gong!!

The solution is fairly simple..

Assume the impulse delivered at the hinge to be zero. Due to the impulse I, the gong acquires an angular velocity w..

Newton's law abt. the hinge gives:
I(x)=(MK^2+MR^2)w.........(1).


Note that
MK^2 is the moment of inertia abt. the COM.
I(x) is the angular impulse abt. the hinge.

The same abt. the center of the gong gives:
I(x-R)=(MK^2)w..........(2)

Solve these to get x = [1+(K^2/R^2)] times the radius..

Some of you may appreciate the result..i'll post some cool similar questions someday.

Supplement

This should be cake...

A brick is kept on a rough inclined plane. Consider the net normal reactions on part A and B of the brick to be 'N' and 'N*'. (A and B are two equal portions)
Then


a)N > N*
b)N < N*
c)N = N*
d)We need the angle of the plane and coeff. of friction to comment...

Anyone interested?? Plz post only the option..not the soln..
This should be fun!

Stcky Tape

This disc of mass M, Radius R rolls down from a height H without slipping.

The floor below the hill has sticky tape of linear mass density P..and it winds around the disc as it rolls (its sticky side faces up). Finally the disc stops!!!!!

What is the total distance traveled by the disc apart from that traveled down the hill??
assume the tape to be really thin..also assume that the disc travels a sufficiently large distance.

Snakey..

Here's the Soln:

x=9LCosQ...
So dx/dt=-9LSinQ(dQ/dt)

to evaluate dQ/dt use:
y=LSinQ
dy/dt=v=LCosQ(dQ/dt)

So dx/dt=9LSinQ(v/LCosQ)...so, dx/dt=9v...

Dudes, this is the Standard approach:


1)You write the coordinates of the point in question and those of the pts. whose velocities are given...relative to a fixed pt (the origin).

2)Differentiate these..

Zone Transition 1

This temple has its gong in the shape of a solid sphere..however the hinge (as shown) is really weak..

Specify: Where should the gong be hit by devotees to impart minimum damage to the weak hinge????..in terms of the radius of the gong.

And yaar i'm really busy these days..so i'll post all the solutions some other day..

Aur yaar this one's important and good..
I would like you to solve the same for a rigid body of radius 'R' and radius of gyration 'K'...
For a rod??? you could mail your ans. to Tendulkar..he would love to save his hands from those jerks....

Bounce...

A ball is dropped from a height 1m into a large reservoir of water.

The density of the ball is .5gm/cc...So what is the total distance traveled by the ball in air till it stops??

Finally back!!

So, I'm back from a well earned break..and i really missed posting questions. Sadly, the place I went to has little internet..but it has a good Shivratri Mela..

I had always wanted to analyze this toy (available at the fair) for its Physics. Could you do it for me??

I have assumed that the two ends, one in each hand are moved as shown with some velocity V. With what velocity does the snake-head move when the situation is as shown (ie the angle is 90*)???

All segments are of equal length 'L'..