I was watching TV at odd hour asusual. And stumbled upon this interesting bit of program
This is what they showed first. An ice skater (a kid) started spinning by applying force and after spinning with a reasonable speed for few seconds by keeping her one leg on the ground and other perpendicular to it (can you imagine it?) she started shifting her position to standing straight and started spinning really fast without applying any extra force. (I am sure the visualization would have made this whole thing more interesting). I was surprised to see her spin so fast and thought this is some kind of trick she is doing. But then a person asked her how she is doing that and asked her to stop since she is violating the newton’s 2nd law of motion. (A body at motion will be going at same speed unless an external force acted upon it)
This kid said she learnt to do it and not sure about any physics behind it. Then the narrator (I assume she is some kind of techie) starts explaining step by step.I am trying my best to reproduce it before it evaporates from my brain.
Momentum works only for body in straight line. For bodies rotating it is called angular momentum. Where momentum depends on Mass, angular momentum depends on I (angular inertia). Angular inertia depends on Mass and Shape of the body. In case of disc (CD), the mass is equally distributed and distance from the center to end of the object is R (radius). So the totally angular momentum is 1/2mR2 (read as half m r square).
The inertia depends on axis of rotation too. For example, a cylinder spinning with a axis going through its center (center of circle) will have less inertia than the cylinder spinning in its length axis. vs --- . For the first case the I = ½ m r^2 and I = 1/12 m l ^2
I = ½ (40kg) (.15m)^2 = 0.45 kgm2 (M = 40Kg; her radius/width = .15m)
I = 1/12 (40kg) (1.2m)^2 = 4.8 kg m2 (M=40kg; her length = 1.2m)
That explains why her inertia is more when she was spinning with her legs in horizontal position and why it is less when she is spinning in upright position. That in turn explains why she could spin faster when she is in upright position
But still that doesn’t explain how she could increase her speed when she shifted from horizontal to vertical without applying any extra pressure.
She continued. According to law of conservation of momentum, momentum should be maintained somehow. And her is the formula for it
L (angular momentum) = I (angular interia) * w (angular velocity)
Since she shifted from horizontal to vertical her inertia changed from 4.8 to .45 and only way to maintain/conserve the angular momentum is by increasing the angular velocity so her velocity went up.
WOW! Sometimes it useful to stay late (1.45AM) to learn physics, other times you may find interesting “Magic bullet” advertisements.
Have a good new year!
Monday, January 02, 2006
Physics at 1.45AM
Posted by BrainWaves at 2:25 AM
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15 comments:
Very interesting blog... It made me go back to basics on what angular momentum is..I had my own questions though... My ultimate conclusion is that, newton's law is probably not violated...
These was my first question. Is it the radius "r" that matters or "r*sin-theta" since angular momentum is mvr*sin-theta. If radius determines the inertia, let us assume that the skater holds a stick pointing outward during the start and then brings it back vertically, will her speed increase? Or, if she holds an umbrella open and then close it suddenly, will it increase her speed? I dont know.
Then, I started thinking about the speed itself. The toe of her leg(horizontal) has to cover a larger distance than the toe in the ground (of the other leg). So, if speed is distance covered by the toe of the leg pointing outward, it will be a bigger circle. When she brings her toe back to her feet (vertical), she still has to maintain the same speed and hence she has to rotate for the same distance, which would mean more rotations. Remember, the speed is the same. So, it will appear as if she is rotating fast, when in reality, she actually might be covering the same distance. I might be missing something, which I will investigate.
Shoba deserves to be mentioned in participating in discussions related to this...:)...
Well. I was eager to put things on computer asap and to watch Dhoom (after typing that blog) so missed few details.
Interia depends on the weight/mass of the object rotating and how it is "distrubted across its axis".
So, incase of a pole if the weight of the pole is negligible then it does not matter. But if the weight of the pole is high and if she put so much power to make it rotate along with her, then according to the definition she will rotate faster when the pole is vertical.
It was intereting to watch the kid rotate so fast without putting any extra effort.
Shoba you rock :)
Brainwaves,
The dependence of inertia on mass clears some questions. I have an important question,though. Did you write down these equations in the middle of the night. If you had remembered them off-hand at 1:45 AM, you must have been really wide awake..
Brainwaves, I think we are on the same page. My explanation in the second paragraph is just another way of defining inertia, after all. One key thing we should note is that, the angular velocity is distributed along the length of the leg and is highest at the edge (assuming mass per slice of leg is uniform). For the picky me, your second equation that calculates 4.8kgm2 should use the mass of the leg alone and not the entire mass of the body (infact, it should be an integration). Also, if they hold their hands out and then retract it(along with legs), then, they should get even more rotations per second. Infact, the easy way to calculate the number of extra rotations per second is nothing but ratio of r2 divided by r1 (approximately) since the distance travelled will move from 2*pi*r2 to 2*pi*r1.
One more thing that I forgot to mention... Where did theta go? In this case, theta=90 degrees, so sin(theta)=1 and hence we dont see it. Otherwise, we would have to include that as well...:)... I dont know what it did to you when you were awake, but, to the sleeping me, it woke me up...:)... We should have more of these kind of discussions....
She was not only extending her legs out but also leaning to one side so the weight is all but one leg which was pointing downward.
With respect to equation. I typed it as soon as I heard the program so it was good reproduction.
Leaning doesnt make all the mass to be at distance "r". The m*r^2 holds good only for mass m at distance r from the axis of rotation, not the mass of the entire body... thats my understanding...
Consider this case. Let us assume the skater has an object of mass "m" tied by a weightless string and she is also of mass "m" trying to swirl the object around her. The inertia is still mr^2 (assuming her radius itself is negligible) and not 2m*r^2. Hope it clarifies it.
Hmmm...thinking about it, the only way the equation would make sense is if she started with almost a T position with a leg on one side and her entire body on the other side lying in straight line with another leg stuck to ground... In this case, we can approximate the inertia to be due to the entire mass "m"...phew...:)
well that is what I meant. Where is the picture attachment option when we badly need one :)
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Reason for tilting one side is to compensate the centrifugal force acting on the body.
Yes. Ballerina tilting will slow her down but that is due to increased inertia. (equivalence of friction in a straight line object)
To continue this technical discussion, I wanted to tell u guys that I recently found out. These questions will especially be more interesting for Calm & Serene's mother.
1) We all know what derivative of a function is. When we take derivative of a derivative, we represent it by d^2y/dx^2. What is d^0.5y/dx^0.5 (half derivative) for the function, say y = x^2?
2) If you try to find the solution to the previous problem, you will hit upon the following question...(1/2)! = ? (In general, what is fractional factorials and how to find the value of those?)
I tried the first part just using the generalised nth order derivative of x^n and substituted the value of n=1/2 in the answer. If u do the same, you will find the fractional factorial term in the derived equation and apparently, it can be found using "gamma function".
I've never thought about 1/2 derivative and it was interesting to derive these equations to refresh my calculus memories. Hopefully, it will refresh you guys as well...
PS: Richard Feynman apparently solved this problem numerically when he was 12 years old...
the reason bike/people tilt in turns is the same reason roads are banked in a turn.
if a car made a flat turn at sufficient speed it will slide right off. traction is a function of weight on tires. banking alows some of the g-force created in the turn (horrizontal) to increase the weight on the wheels by aligning the wheel along the horizontal plane (if not completely atleast a portion to get a fraction of the horrizontal G).
Same reason for a bike to bank too. if the road is flat- the bike tilts to convert a portion of the horizontal G-force into more weight on the tires.
This is also the reason, it is recommended to accelerate slightly while making the peak of the turn- it increases traction (assuming you were not at the edge of the tires theoretical traction for given tilt). ofcourse above recommendation is more accurate for bikes.
ps: when i use g-force i'm not talking about the gravitational force. but about g-force as a unit of measurement. here it is the centrifugal force that is tangential to the curve and on the horizontal plane.
Reminds me of the banking of rails. One aspect of the problem is the centrifugal force acting towards the center (g-force). The other aspect is that, when a vehicle takes a turn, its outer pair of wheels (on a left turn, it would be the 2 tires/wheels on the right side, both front and back) have to travel a greater distance than the inner side and hence different rpms. In automobiles, there is a differential mechanism which makes the wheels on either side to rotate at different speeds (in race courses, the track itself is banked to make it easier). In trains, however, there is no differential mechanism. The wheel itself has different thickness and it slides along the rail during a turn such that the wider portion of the rail engages with the grove to maintain the different rpms across a curve (imagine a wheel which is shaped like a cone with its sharp portion chopped off)....
u mean centrepetal force acting towards the center. The centrefugal force acts along the tanget of the circle .. or am i getting it reversed.
i remember my first introduction to diferential in the driven wheels of the car (the non-driven wheels don't need a differential since they are free to rotate at diff speeds), when i built my first lego car with connected wheels and during turns it used to feel odd.
Also I suspect along same lines- only the engine wheels need to solve the problem- i suppose the carriage wheels are free to rotate at different speeds since they are not interlinked.
From what you say- the difference in thickness of the wheel along the cone should be designed such that it can tackle the min and max radius of turns that the train will potentially meet. Also it should have enuff play in the wheels to allow the wheels to slide?
Yes, I meant the centripetal force. I was going to correct it after I posted the reply. And then, I was lazy..:)
You are right about the fact that, only the engine wheels need differentials. In a car, based on front-wheel drive or rear-wheel drive, the differential will drive those corresponding wheels. For an all wheel drive, I guess there should be a front-differential and a back-differential and somehow, those two should be coordinated as well...
Yes, the min-max radius of the engine wheels (in the case of train) should be designed based on the maximum steepness in curve that the train will see.
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