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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 ...


65

The Foucault pendulum is a great experiment which does demonstrate that the Earth is rotating, but it was only introduced in 1851. The Earth had been known to rotate for several centuries before that, probably stimulated by Copernicus and Galileo pushing the heliocentric model of the solar system during the 16th century. A couple of decades before Faucalt's ...


49

Foucault pendulum. I don't know how the ancients did it, but it is surely pure classical mechanics. The animation describes the motion of a Foucault Pendulum at a latitude of 30°N.


48

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 ...


21

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 ...


16

I think the Foucault pendulum is the best answer, but for the sake of variety I'll add another very interesting one: the equatorial bulge affecting the figure of the Earth. This is the "pancaking" of the planet due to its rotation. You can measure the geometry of the Earth without leaving its surface, and find that it is bulging in accord with your ...


13

From conservation of angular momentum we have $(I+\Delta I)(\omega+\Delta \omega) = I\omega,$ or $$\frac{\Delta \omega}{\omega} = - \frac{\Delta I}{I+\Delta I} \simeq -\frac{\Delta I}{I}.$$ We make the following simplifying assumptions: The earth is a sphere of uniform density of mass $M$ and radius $R$. The building is constructed on the equator by ...


12

An indirect indication that the Earth rotates is the fact that the rotation varies over time. First of all, the orientation of the Earth's axis changes: long-term effects like precession and slow variations in the axial tilt, as well as small short-term variations like nutation. Precession was already known in the Ancient world (Hipparchus, Ptolemy,...) and ...


11

The system needs to conserve momentum. In both cases, the momentum is whatever m*v is for the bullet. Since it's the same in both cases, the bullet and block have the same vertical velocity. Mechanical energy is not conserved. The reason the block hit on the side has more kinetic energy is that the bullet converted less of its kinetic energy into heat upon ...


10

Well, if we make a quick estimate of the mass of a huge building. Let's say the building has a base of $100\times100 \;\text{m}^2$ and a height of $1500 \;\text{m}$, this is already substantially bigger than the current biggest building. Then we have a volume of $1.5\times 10^7\text{m}^3$. If we make the assumption, again very rough and on the high side, ...


9

The moment-of-inertia (MOI) tensor is real (no imaginary terms), symmetric, and positive-definite. Linear algebra tells us that for any (3x3) matrix that has those three properties, there's always a set of three perpendicular axes such that the MOI tensor can be expressed as a diagonal tensor in the basis of those axes. These are called the principal axes ...


8

ZPM isn't the full answer. A combination of gyro and thrusters are used. Primarily they use Control Moment Gyroscopes (CMG) located in the Unity Module. Secondary options with more thrust are the Russian Control Thrusters on both Progress and Zvezda (means star) modules. The CMG's are quite heavy gyroscopes at about 600 lbs each. Inside the black ...


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

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

The rectangular prism is a rigid body. The equations of motion of a rigid body around its center of mass are given by: (Please, see for example: Marsden and Ratiu , (page 6). $$I_1\dot\Omega_1=(I_2-I_3)\Omega_2\Omega_3$$ $$I_2\dot\Omega_2=(I_3-I_1)\Omega_3\Omega_1$$ $$I_3\dot\Omega_3=(I_1-I_2)\Omega_1\Omega_2$$ Where $\Omega_1,_2,_3$ are the angular ...


7

Here are some supporting evidence, taken from here. The inner core rotates in the same direction as the Earth and slightly faster, completing its once-a-day rotation about two-thirds of a second faster than the entire Earth. Over the past 100 years that extra speed has gained the core a quarter-turn on the planet as a whole, the scientists found. Such ...


6

The instability inherent in the medium length axis or $\prod_2 $ as shown above is discussed in detail in Marsden and Ratiu, which is where the image is from. The unstable homoclinic orbit that connect the two unstable points have intersting features. Not only are they interesting because of the chaotic solutions via the Poincare-Melnikov method that ...


5

You can always decompose a motion like this into two parts: (1) rolling without slipping and (2) slipping without rolling. What is slipping without rolling? It means the object moves uniformly in one direction along the surface, with no angular velocity about the object's own center of mass. For instance, a box that is pushed along the ground can easily ...


5

As in the answer of @Mark Eichenlaub the mass of the lawnmower won't increase! Of course the blades of the lawnmower can have a pull-effect in which they might aid your foreward-movement (in which degree this might help, I'm not sure ...). Of course a rotating blade creates an angular momentum, so if you were to make a turn with your lawnmower you'll need ...


5

If you start in the rest frame of the wheel the velocities of the top and bottom points are $v$ and $-v$, and the velocity of the centre of mass (black dot) is of course zero because that's how we define the rest frame. If this wheel is on a moving vehicle the velocity of the bottom must be zero, because it's in contact with the stationary road. To make ...


5

I have observed this as well, and experiment suggests it's because the dust is hydrophobic. If you splash a small amount of water gently onto the dusty surface you will see the water roll up to form beads that do not wet the surface. This is my rather crude attempt to illustrate what happens when you try and wet the dusty surface: The brown splodges are ...


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

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 ...


5

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 = ...


5

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 ...


4

The electric field is nonzero. For a cylinder of finite length, it's nonvanishing everywhere. In the limiting case of an infinitely long cylinder, the field is only nonvanishing inside the cylinder. The easiest way to solve this is to use the fact that the electric and magnetic polarizations $(-\textbf{P},\textbf{M})$ transform in exactly the same way as ...


4

The angular momentum of each disk individually is not conserved, however the total angular momentum of both disks is conserved because there are no external torques acting. Start by calculating the total angular momentum of both disks (I'm going to replace "w" by "v" since "w" is confusingly close to "$\omega$"): $$\begin{align} L_{total} &= I_a ...


4

Measuring the geometry of the earth, we find that it has an equatorial bulge. We make no assumptions about the cause of the bulge, though it suggests already that the earth is rotating as @Mike has described. We measure the acceleration due to gravity at the poles and on the equator. Most of the difference we find is accounted for by the bulge, but there ...


4

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 ...


4

The direction of angular velocity is different from that of regular velocity for (arguably) two reasons. First, it points out of the plane because of the nature of angular velocity. It signifies a rotation, as such, there is not any particular direction unit vector in every coordinate space that could represent it. In spherical or cylindrical coordinates, it ...



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