Cubic resistor

A wire-frame cube is made of 12 resistors of resistance R. What is the resistance between diagonally opposite corners of the cube?

When the wind blows

A cyclist cycled a mile in 4 minutes against the wind. He returned in 3 minutes. How long would it have taken if there was no wind?

Up and away

Two balls of greatly different masses are dropped (almost touching) from a height of 1 metre. They fall through the same path, with the light ball following the heavy ball. After the bounce(s), how high will the lighter ball get to?

If you were able to cascade a series of balls together (with each upper ball being much lighter than the one below it), how many balls would be needed for the uppermost (lightest) ball to achieve escape velocity? Assume that escape velocity = 11 km/sec.

Assume that the balls have neglible diameter compared to 1 metre. Neglect air resistance and only make simplifying assumptions.

Two Cars and a Bird

Two cars, setting out from points A and B are 140 miles apart. They move toward each other on the same perfectly straight road, until they collide at C. Their speeds are 30 mph and 40 mph. At the very instant they start, a bird takes flight from point A heading straight toward the car which has left point B. As soon as the bird reaches the other car, it turns and changes direction. The bird flies back and forth between the two cars at a speed of 50 mph until the cars collide.

How long is the birds total flight path?

Ice bound

A glass of water has an ice cube floating in it. When the ice melts, will the water level rise, fall or remain the same?

Please ignore evaporation and other such sophistication.

Bouncing ball

I've just found this. I haven't got an official answer, but here goes:

A ball is dropped from a height of 1 m on to a surface. During the bounce, the ball loses half of it's energy. How long does the ball bounce for?

Assume everything else is ideal.

(I know Ragknot will have a field day with this.)

Hole in the Sphere

A cylindrical hole six inches long has been drilled straight through the center of a solid sphere. What is the volume of the solid material remaining in the sphere?

Bicycle Pedal

A wire is tied to a cycle pedal that is stationary at the bottom of its arc. If someone pulls this wire backwards (while another person lightly holds the seat to keep the bike balanced), will the cycle move forward, backward or not at all?

Give reasons !

About Arrows

What makes arrows fly straight ?

Candle Problem

You are in a dark room and seeing a lighted candle at 6-to-8 meters distance. Slowly move towards it till the candle is right in front of your eyes (a mm or two away). As you move towards the candle the eye stars accommodating to keep it in focus. This it can do to a certain maximum point beyond which the flame goes out of focus. Since the candle is still moving closer to the eye, its focus which till now was on the retina will keep going backwards, so much so that an inverted image formed on the retina should at one point become upright. Then why don’t we see objects which are very close to the eye, upside down?

WARNING: DON'T TRY THIS AT HOME, YOU CAN BURN YOUR EYE BROWS OR WORSE LOSE YOUR SIGHT

Galileo’s Twist

Climb up the Eiffel tower, a rod made of one metre iron and one metre wood. Holding it horizontal drop it. It will reach the ground with the heavier iron side touching down first. Is this a violation of Galileo’s: time of fall or acceleration is independent of mass?

Space Elevators

Armstrong says space elevators must be built. He’s planning to build a carbon nanotube ribbon that is three feet wide, anchored somewhere in Ecuador, stretching a mind-numbing 62,000 miles upwards into space. The centripetal force of Earth’s rotation will keep the ribbon taut. The problem is – if I sneak on to the platform in Ecuador and snip off the ribbon at its base, what will happen?

Defective mass?

Two carefully prepared specimens of two different materials were manufactured and actually had the same mass as each other (within a negligible error). When they were weighed against each other on a suitable high accuracy pan balance, they didn't balance! Why?

Assume the specimens are solid solids!

Gyroscopes

When we spin a top, initially it wobbles. But on most occasions it straightens up after some time and the spins steadily till it loses its speed and finally falls. Why does it become steady after the initial wobbling?

Stargate

A stargate is composed of 39 chevrons. Locking in 7 chevrons establishes a wormhole to a location within our galaxy. Locking in 8 chevrons establish a wormhole to a different galaxy. Assuming that 9 chevons would link to alternate universe, how many universes would that be?

Galileo's Principle

Air is a vapor (gas), which means that its atoms and molecules are not in such close proximity to each other as in a solid and liquid. So why don't air molecules just fall to the ground? After all, Galileo's principle states that objects should fall to the ground with equal acceleration independently of their size and mass.

Floating hour glass

This is an old mystery. People have argued about the solution. It was published in Omni Magazine back in the '60. You can see it and read about it. But there's little argeement about the "why".

An hour glass containing air and sand is in a cylinder of water. While sand is flowing from the top to the bottom compartment, the hour glass remains submerged at the bottom of the cylinder, but when the sand has completed its downward journey, the hour glass floats to the top. Why?

Two abreast

This is probably my last dabble with relativity (for a while).

Prove that if in a given inertial frame, two photons are travelling in parallel and abreast, at a distance r apart, then they do so in all inertial frames.

Hints: Consider the case when they are not travelling abreast and then when they are abreast. There is a nice way to do this.

Link to the reference book for this problem:

here

Star Trek

I was researching for a new problem, but decided that the following link was more interesting: http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html

OK, I've thought of a question: how fast must the rocket be going in order that the microwave background temperature rises to 10000 Kelvin? Ignore GR considerations, treat it as a SR doppler shift problem, use simple black-body radiation concepts only (unless you can do better).

Phasers on stun

Sorry, this is a bit advanced - I won't do too many like this.

In the inertial rest frame of a dust cloud of identical particles, at some time t'=0, all the dust particles spontaneously and simultaneously begin to emit light.

What physically happens in an inertial frame where the dust is moving with speed v?

Compare this with the phase velocity of the associated de Broglie wave.

On the rebound

A cubic block of metal of mass M is hanging from a long thread. You throw a perfectly elastic rubber ball of mass m at speed u, perpendicularly to one of the block's vertical faces. Immediately after the ball bounces off the block, what is the speed of the ball and the block?

Unbalanced balance

You have a balance whose fulcrum is not at the centre of the pans. When you put a mass m on the left hand pan, it is balanced with a mass m1 on the right pan. When you put the mass m in the right hand pan, it is balanced with a mass m2 on the left hand pan. What is m?

Rain with an Angle

Why is there less rain per unit area in 45degree tilt?

Conveyor belt

A conveyor belt has speed v m/s. A stream of goods is being loaded onto it via a hopper, at a steady rate of m kg/sec. How much power has to be supplied to the conveyor belt to compensate for the load?

Neglect any unnecessary complications.

Sorry, a bit boring.

Hold on

A particle is in an upright circular path of radius r. At what speed, u, must it be injected (at the bottom) in order that it can go right round the track without falling off?

Neglect friction, drag and variation of g with height. Assume that the mass slides (it doesn't roll).

-

The time it takes to go round the loop involves non-analytic elliptic integrals. I can do it but only thanks to wolframalpha.com

Mixing it

You have jugs A and B that contain the same amount of differing liquids. Take a small sample from A and mix it into B. Then take the same size sample from the A+B mix and mix it into A. What is the relative concentration of A in B to B in A?

There's more than one way of solving this.

Bug on a Rubber Band

An infinitely long rubber band has one end nailed to a wall, while the other end is pulled away from the wall at the rate of 1 m/s; initially the band is 1 meter long. A bug on the rubber band, initially near the wall end, is crawling toward the other end at the rate of 0.001 cm/s. Will the bug ever reach the other end? If so, when?

Dishes and Ratings

A fast food place has N dishes on its menu that are rated from worst to best, 1 to N. You, however, don't know the ratings of the dishes, and when you try a new dish, all you learn is whether it is the best (highest rated) dish you have tried so far, or not. Each time you eat a meal at the restaurant you either order a new dish or you order the best dish you have tried so far.

How can you maximize the average total ratings of the dishes you eat in M meals (where M is less than or equal to N).

Beam me up

You have two *square-section beams of wood. One has a density greater than half that of water, the other less than half that of water. You throw them in a pond. What do you notice about the orientation of the floating beams?

*In the original posting, I unintentionally wrote "rectangular".

Take the pole into the room

The pole from the previous flatland post, is now going
into a large room off a hallway. The walls are 6 inches thick,
the door is 30 inches wide. The pole is 24 feet long.
The hall is not very wide. How wide would it have to be for
the 4 inch diameter pole to be taken into the room without bending it?

Hallway Pole

You are an architect. Your client, living in Flatland, wants a building designed with a long 3 foot wide corridor which opens into a larger hallway. You must design the hallway for the minimum width which will allow the inhabitants to move a 24 foot-long pole down the corridor and turn it into the hallway.

The corridor is perpendicular to the hallway. Since this is Flatland, the pole cannot be tilted up. The pole is rigid. How wide must the hallway be?

One Half Pill Problem

You have a prescription to take one half of a pill per day for 12 days, but the pharmacist (who is too busy to divide pills for you) gives you 6 whole pills in a bottle. On day 1, you remove a pill from the bottle, break it into two half-pills, take one, and return the other half-pill to the bottle. On all subsequent days you shake the bottle thoroughly and pour something out - whatever comes out first - either a half pill or a whole pill; if it's a half pill you take it and you're done for that day; if it's a whole pill, you split it into two half-pills, take one, and put the other back in the bottle, exactly like you did on day 1.

On day 12 there can be only one half pill left in the bottle, but on day 11 there are two possibilities: either there is one whole pill or there are two half-pills left in the bottle. What is the probability that there are two half-pills in the bottle on day 11?

Weight Problem

A chain of length A cm and mass B kg is suspended vertically by one end with the other end immediately above a scale. The chain is released and falls. At the instant the entire chain has fallen onto the scale what does the scale read?

Maximum Velocity

A shell flying with velocity 500 m/s bursts into three identical fragments so that the kinetic energy of the system increases 1.5 times. What maximum velocity can one of the fragments obtain?

Mass at Planet51

A space traveler about to leave for Planet 51 has a spring balance and a 1.0 kg mass A, which when hung on the balance on the Earth gives the reading of 9.8 newtons.
Arriving at the Planet51 at a place where the acceleration of gravity is not known exactly but has a value of about 1/6 the acceleration of gravity at the Earth's surface, he picks up a stone B which gives a reading of 9.8 newtons when weighed on the spring balance. He then hangs A and B over a pulley and observes that B falls with an acceleration of 1.2 m s–2 .

What is the mass of stone B?

Patient, Pills, and Probability

A Patient is taking one each of 5 different types of pills every day but he don’t like having to open and close 5 different bottles, so at the beginning of each (30-day) month he put 30 of each type of pill into one big bottle. When it is time to take your pills, he draw them out of the big bottle one at a time until he have (at least) one of each type.

On the last day of the month he will draw exactly 5 pills and they will all be different (because that’s all that’s left in the bottle), but on other days he will generally have to draw more than 5 pills in order to have (at least) one of each type. So, the question is: On the first day of each month (when there are 150 pills in the bottle), how many pills, on average, must he draw from the bottle in order to have (at least) one of each?

Archimedes Principal

A fisherman rowing his boat on a very small lake throws his anchor into the water. Does the water level of the lake rise, fall, or stay the same?

Do you know Gravity ?

If you throw a small ball vertically upward in real air with drag, does it take longer to go up or come down?

Beetle on Sphere

A Beetle starts from rest at the top of a sphere of radius R and slides* on the sphere under the force of gravity. How far below its starting point does it get before flying off the sphere?

* assume sphere is frictionless.

Pile of Bricks

A uniform brick of length L is laid on a smooth horizontal surface. Other equal bricks are now piled on like a ladder , so that the sides form continuous planes, but the ends are offset at each brick by a distance L/a, where a is an integer. How many bricks n can be used in this manner before the pile topples over?

Siphons

How does Siphons works ?

Conversion factor

Ragknot got an email from Lori that said the axpproximate
conversion factor to use was 0.1417322834.

He thought a minute and said it would be exactly 36/254.

Question: What units was Ragknot convert to and from?

Hint: ____ per ______ * 36 / 254 = ____ per ____

The Refrigerator

One day, Dr. Nova arrived in a laboratory to find that his technician had left the refrigerator's door open.

When he pointed this out, that man coolly replied, "It was such a hot day and I wanted to lower the temperature of the room a bit."

Dr. Nova was amused. Why ?

King and the Goldsmith

King had given a goldsmith some gold and asked him to make a wreath from it. When the goldsmith finished the wreath, he suspected the crafty goldsmith had pocketed some of the gold and replaced it with some cheaper metal. Yet the wreath weighed exactly the same as the original gold.

How could the fraud be proven ?

Wood Beam

The strength of a wooden rectangular beam varies jointly as the width and the square of its depth. Which rectangular beam that can be cut from a circular log of radius 10 inches will have maximum strength?

Snowflakes

Shetty, Sameer, Sandeep, and Rajesh planned for a reunion at Minneapolis MN in the weekend. There they come across a unique kind of snowflakes, which according to Sameer, happens only at this time of the year, that too at a very unique temperature and when air flows in a very unique speed.

After returning, Rajesh, still intriqued by the phenomenon called Sameer to ask what was the temperature which caused the Snowflakes of such design and what was the air speed, to which Sameer gave two numbers. When Rajesh enquired whether the temperature was in Fahreheit or Centigrade and the speed kilometer per hour or miles per hour, to his astonishment Sameer said:

"It absolutely makes no difference !"

How ?

- Bob (Robert)

Kinematics

A dirt-biker rode the first half distance of the race with the speed of 200 miles per hour. For the remaining part he was a bit distracted and covered half the time with the velocity 150 miles per hour and rest of the time, speed up to 250 miles per hour.

If he came second and was beaten by 20 seconds. What was the average velocity of the guy, who came first ?

Einstein's Sphere

An Incredible Sphere was found in the laboratory of Einstein. A cylindrical hole, 6 inches long has been drilled through the center of the solid sphere. Remaining volume of the sphere was needed to solve the puzzling equation, written on the black board.

Can you help?

Hot

Is boiling water always hot ? If not, why not ?

Are we there yet???

Less riddle, more paradox:

To go from point "A" to point "B," one must first go half way. When at the half-way mark, there is now a new point "A." To travel to point "B" takes another trip first to the half-way mark. Since this is and will forever be a renewing course of events, how can point "B" ever be reached?

In the same area of thought; If you break a rock in half, what is left? A half-rock? No, of course not. There is no such thing. You have two rocks. Break one of them in half, and yet another smaller rock. Half again, and again, and again......Is this infinity? Physics will wind you up at molocules and atoms. Religion; God. Others will say you will find things called "Super Strings." Steven King and others say galaxies within galaxies.

I think........well, who cares what I think.

What do YOU think?

Notorious Tank Problem

The notorious Tank Problem that one finds in every single collection of arithmetical and algebraic problems. No doubt, you all remember those classic scholastically dry problems of this order:

"The tank has two pipes -- one leading to fill the tank to the brim and the other out. the first needs five hours to fill the tank and the second ten hours to drain it dry. How long will it take the tank to fill up when the stop cocks are out of both pipes ?"




This problem has a venerable history, dating right back some twenty centuries to Heron of Alexandria. Here is one from his collection for your free time in the weekend.

Four Fountains are there and a reservoir vast.
In but one day the first doth fill it to the brim.
The second two days and nights must play to do the same.

The third takes thrice the time as did the first.
The fourth comes last with four days and nights.
Now tell me when the reservoir will fill,
when all four play at once.



It is two thousand years now that the tank problem has posed and -- such is the force of habit ! -- has been solved wrongly in all this time.

Magnetic "Perpetual" Motion

In attempts to invent a "perpetual motion" machine the magnet and its power have played a role of no mean magnitude. Ill-starred "perpetual motion" machine inventors have tried might and main to apply the magnet to this end. Here is one such project described back in 17th century by Englishmen John Wilkins, the Bishop of Chester.

A Powerful magnet A is placed on the top of the pillar leaned against which is two inclined grooves M and N, one above the other. The upper groove M has a small hole C at the top, while the lower groove N is curved.



The inventor claimed that the arrangement would operate as follows. A small iron ball B was to be placed on the upper groove. Attracted by the magnet A, it ought to roll upwards. On reaching the hole, it should roll down, be carried up by inertial along the curve D, and find itself again on the upper groove M, from whence, again attracted by the magnet, it should again roll up and drop through the hole, roll down and on to the upper groove, ad infinitum.

This the inventor conjectured, would produce "perpetual motion".

What do you think ? Will it ?

Why does Rocket Go Up ?

Professor Feynman was teaching the class.
"Even from students of physics one may often hear a totally wrong explanation of a rocket's flight. They claim that it goes up by thrusting itself away from the air with the help of gases formed from gunpowder combustion. That incidentally is what ancients thought-rockets are invented long, long ago."

He added "But if we were to fire a rocket in an airless void it would fly, and even better than in the air."

What's the right explanation ?

Fantasy & Science Fiction

Isaac Asimov had a 30 years correspondence with Nobel Prize winner Linus Pauling. Pauling read Isaac's science articles in Fantasy & Science Fiction magazine regularly and wrote Isaac whenever he found an error. Here is an interesting excerpt.
From Pauling to Isaac:

I am writing now about your article in the September 1978 issue. On page 123, you say that Amontons and Guy-Lusac observed that if a gas at the freezing point of water, 0 degree Celcius, is decreased in temperature to -1 degree Celcius, then both the volume and the pressure of the gas will decline by 1/274 of the temperature. This is wrong. What you should have said is that _ _ _ I hope you are keeping busy as ever.

What was the explaination?

A Motorless Underground Railway

If you have read the free travel, you will find this more intriguing. A booklet with the odd title A Motor less Underground, Leningrad-Moscow Railway was once put out. Its author A.A. Rodnykh, suggested a very interesting project .

His idea was to "dig a 600-km-long tunnel, linking up two cities Leningrad and Moscow by an absolutely straight underground line. this could give us for the first time the opportunity to travel along the straight instead of following a curved path as now." He wishes to say that all our roads describes arcs, as they follow the curve of the earth surface, while the suggested tunnel will follow a straight line along a chord (assume earth as a circle).

This project - if ever realized - would possess a unique characteristic. In this tunnel a train move by itself. At each point, since the train is going in the downward direction due to gravity, it will not require a motor. The big tunnel will actually be inclined to some angle with the horizontal ground at both the points.

In such a slanting tunnel every object should swing, due to gravity, to and fro like a pendulum, hugging to the bottom. Inside it a train would move by itself along the rails, its weight doing the work of locomotive. At first the train will move very slowly but with every new second its speed would increase, to reach soon a figure so incredible,that the air in the tunnel would offer a noticeable resistance.

Do you see any flaw in this design, A motorless vehicle can help realize the dream of free travel. Why this is not done yet?

Fatal Attraction !

In the autumn of 1912, the ocean linear Olympic, one of the world's biggest ships at the time, was steaming ahead out in the high seas, when another much smaller ship, the cruiser Hawk, rapidly approached it on a parallel course a hundred meters away.

As soon as the two ships took up a position where one pointed the other, a surprising thing happened. the Hawk sharply veered off its course, as if obeying some invisible force, turned the big liner and heedless of the helm rammed into it.The impact was so great that it made a big gash in the Olympic's hull.

A tribunal examined this queer case and found the Olympic's Skipper guilty, as, according to its ruling, he had failed to issue orders to yield the right of passage to Hawk. Consequently, the tribunal, as you can gather, saw nothing extra ordinary about it at all, attributing the the accident to the skipper's negligence.

Actually this was the result of a totally unforeseen cicumstance, a case of mutual attractions of ships at sea and the answer lies in basic physics.

Why do ships attract one another ?