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69

In comparing wheels of today to those in history, there are traditionally more spokes now. However, that's because wheels in the past (even large wagon wheels in not-so-ancient times) used relatively thick wooden spokes that behaved like a column and dealt with the load of the wheel with compression. However, modern spokes are very thin. Far too thin to ...


56

The astronaut can change his or her orientation in the same way that a cat does so whilst falling through the air. After the transformation, the astronaut is still and angular momentum is conserved. There is a rather beautiful way of understanding this rotation as an anholonomy i.e. a nontrivial transformation wrought by the parallel transport of the cat's ...


52

Your intuition about spinning fluids is wrong for a couple reasons. Angular momentum is conserved so an isolated system of any shape will keep on spinning unless it has a way to transfer that momentum elsewhere. If you spun in egg levitating in a vacuum it would spin forever. The more bumps, flaws, or non-spherical features your container has the faster ...


25

For those that are cat-challenged, here's an alternative explanation and demonstration you can try at home! This demonstration was taught to me by my math lecturer. All you will need is: A swivel chair and a heavy object (e.g. a big textbook) Stand on the seat of the chair (watch your balance now) holding the heavy object. Extend your arms forward ...


22

The Wikipedia article you linked states: Atomic clocks show that a modern day is longer by about 1.7 milliseconds than a century ago If we take this change of 1.7 ms/century and multiply by 2.5 million centuries (250 million years) then we get a change of 4,250 seconds or 1.18 hours. So 250 million years ago the day length would have been 22.82 hours. ...


13

In your comparison with raw eggs and milk cartons, the objects (and the liquid) inside are already at rest and you apply energy to rotate them. However, the entire earth is already rotating with comparably small external torques trying to slow it down. Back to your example, once you get the eggs spinning and try to stop them, they will continue to spin as ...


11

Not sure if I can add much to Kyle's comment, but I'll try. Looking closely, he starts with no angular momentum about the vertical axis - the take-off is "straight". Then he moves one arm behind himself and then extends it sideways - generating a torque about the vertical axis. By tucking his other arm in tightly, his body can now rotate. At the end of the ...


10

One of my favorite scientific papers of all times (mainly because it's rather bizarre) explains the basics of what's going on here. That paper is Kane & Scher, "A dynamical explanation of the falling cat phenomenon," International Journal of Solids and Structures 5.7 (1969): 663-666. To get even more mathematical, there's Montgomery, "Gauge theory of the ...


8

Actually, seismic evidence indicates that Earth's liquid core rotates a little faster than the crust does. This is probably because the rotation-slowing effects of tides with the moon act more strongly on the surface than on the core. Jupiter, the other gas giant planets, and the Sun are all entirely fluid, are roughly the same age as Earth, and rotate. The ...


8

If the ladder is slipping on the floor as well as the wall, then the point of rotation is where the two normal forces intersect. This comes from the fact that reaction forces must pass through the instant center of motion, or they would do work. In the diagram below forces are red and velocities blue. If the ladder rotated by any other point other than S ...


8

I think the solution has more to do with the tennis racket effect (see: http://physics.stackexchange.com/a/17507/392). Let me clarify the disk with hole in it has two stable axes of rotation and one unstable one. The unstable one is through the hole and the stable one is across (below in green) and normal to the disk. I have confirmed that without ...


8

Yes. It turns out that your $T_L$ is equal to $-T/\omega$, where $\omega$ is the angular velocity and $T$ is the usual temperature. We normally work with the reciprocals of such quantities, and in the language of non-equilibrium thermodynamics we say that a gradient in $-\omega/T$ is the "thermodynamic force conjugate to" a flow of angular momentum. Within ...


7

First of all it is a bit strange to express the change of Earths rotation in miles per second every 100 years, since the speed due to Earths rotation depends on your position on Earth. It would be better to express it as an angular deceleration, so for example in radians per second squared. But lets assume you mean the velocity at Earths equator, which has ...


7

Neither article that is quoted shows "$4.7 \cdot 10^{-4}$ miles per second". The Wikipedia article claims that a day grows longer by about $1.7$ milliseconds per century, that is say $86,400.0017$ instead of $86,400.0000$ seconds. Around the equator, the distance covered in a day is exactly $40,000$ Km (that's how the kilometre was initially defined). ...


6

I think I initially misunderstood the question. I now believe there are three components here: a rigid massless rod in the shape of a spiral; a massless spring that is wound around the rod in a frictionless manner; and a bead at the end of the spring. The entire system is anchored at the point A by a single nail - thus, the system is free to rotate about ...


6

Surpringingly the top speed is not necessarily anything to do with friction, though friction will of course have some effect. A motor acts as a generator, i.e. if you turn a motor it will generate a potential difference just like a generator, and this potential difference (usually called the back EMF) is proportional to the motor speed. So if you connect a ...


6

Backspin! Those shots in which the cue ball "draws" backwards after hitting the target ball involve backspin. Without backspin, the cue ball cannot reverse direction. Consider what happens when the cue ball is not spinning at all when it hits the target ball. The cue ball will come to a dead stop if it hits the target ball straight on. Think of Newton's ...


6

The moment of inertia is a rank 2 tensor not a scalar. You'll commonly see it written as a scalar, but this is because by choosing your axes to line up with the principal axes of the object the matrix representing the moment of inertia can be diagonalised: $$ {\bf I} = \left( \begin{matrix} I_{00} & 0 & 0 \\ 0 & I_{11} & 0 \\ 0 & 0 ...


6

The core of the earth is solid !!!!!!! And another one Crust The crust is the thin, solid, outermost layer of the Earth. The crust is thinnest beneath the oceans, averaging only 5 kilometers thick, and thickest beneath large mountain ranges. Continental crust (the crust that makes up the continents, of course!) is much more variable in thickness ...


6

Suppose the ramp wasn't there, then the trajectory of the object would the same as if it fell off a cliff: To get the equation of motion you simply note that the horizontal and vertical coordinates are given by (neglecting air resistance): $$ x = ut $$ $$ y = \tfrac{1}{2} g t^2 $$ So you can get the trajectory by substituting for $t$ to get: $$ y = ...


6

There's another way to do this also, more akin to how spacecraft actually do it: Take a weight on a string, hold it up and spin it. You'll turn in the opposite direction. When you stop it you also stop turning. Of course this will produce an off-axis force that will be a real pain to deal with. Real spacecraft do it by means of a set of internal wheels ...


5

Other answers have pointed out other ways that might be more efficient, but one very simple way to do it is as follows: start with both arms parallel to the body. Then swing them both backward, up over the head, and then back down in front of the body, leaving them back in the starting position. After this manoeuvre, the body will be oriented in a slightly ...


5

If there is weight on the axle the rim gets pushed down into the ground and tries to deform by flattening on the bottom and bulging right besides the ground. Properly tensioned spokes will counteract this bulging and lessen the deformation allowing for an easier and smoother ride. This means that the rim does not have to be super resistant to deformation ...


5

Because the moment of inertia for a point mass is: $$ I = mr^2 $$ When calculating the moment of inertia for continuous bodies we use calculus to build them up from infinitesimal mass elements, so effectively to calculate the moment of inertia of the disk (without hole) we're doing: $$ I_{disk} = \sum_i^{disk} m_ir^2 $$ for the collection of ...


5

The length of the arc PQ is $r\Delta\theta$, as Feynman says, but the difference from $\tan\Delta\theta$ or $2\tan (\Delta \theta/2)$ or something else is negligible because $\Delta \theta$ is assumed to be infinitesimal (infinitely small) in the argument, anyway. For that reason, both angles OPQ and OQP should be taken to be 90 degrees.


5

First of all, if the collision is elastic, the distribution of momentum in between the components is completely determined by momentum and energy conservation! This statement is most obvious in the center-of-mass frame where the total momentum is zero and the two objects are moving in opposite directions. The momentum conservation (the total momentum is ...


5

The Earth's rotation rate and the location of the rotation axis change over time. These collectively are called the Earth orientation parameters. On very short time scales, a day or less, the changes in the Earth orientation parameters result predominantly because of the ocean tides. On the scale of decades to a century or so, the dominant driver is exchange ...


5

(The following answer was written to address the original version of this question, which was simply "Does General Relativity theory correctly explain the ellipsoidal shape of the earth?") Yes. General relativity predicts that the equator will bulge out just enough such that the reduction in gravitational time dilatation at the equator relative to the ...


4

There are two and possibly three factors at work here: Factor 1: Nonscalar Inertia Matrix: Angular Momentum and Velocity have Generally Different Directions One which I don't think has been mentioned is that unless a body is rotating about a so-called principal axis, in general the angular momentum and the angular velocity vectors do not point in the same ...


4

You seem to be saying that friction couldn't speed it up, because nothing else is moving that fast. Well, how fast is it moving? We can imagine the gyroscope axis parallel to the z axis, and the casing to be aligned such that the x axis goes through it. If the casing is tipped slightly, the gyroscope resists that turning and one side of the shaft has firm ...



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