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2

You can think about this in two different ways. One way is to look at the initial and final angular momentum. If you go from $L\cdot(0,1, 0)$ to $L\cdot(1,0,0)$ you need to remove the $Y$ component and add the $X$ component. If you just calculate the difference in the angular momentum, then you get $$\Delta L = L\cdot (1,-1,1)$$ which would immediately ...

1

No. As you realise, this will increase the moment of inertia, which will reduce angular acceleration since the torque which the motor can supply is limited. However, the maximum speed which the motor can reach is not affected, since this depends (mainly) on the aerodynamic force, which is the same - it depends on the shape and size of fan, but not its mass....

2

Displacing some mass closer to the axis of rotation reduces the moment of inertia $I$. Considering Earth as an isolated system (which is not), its angular momentum $L$ must be conserved: $$L=I\omega = \text{const}$$ Therefore if $I$ goes down, the rotation frequency $\omega$ must increase. But if we should also consider the reduction of $I$ while the tree ...

2

I would think that the whole atmosphere surrounding the earth is far heavier than the trees that were cut. The atmosphere turns with the earth and the changed position of trees would not even be noticed.

2

You might hear the story about figure skating. When a rotating person expands his/her arm, he/she can slow down rotation. Same thing can happen with earth. Assuming the tree is trillion tons and you cut it and lift it up, you can slow down the earth.

25

Model the tree as a point mass $m$ located some height $h$ above the ground --- that is, forget the mass of the trunk and assume all the mass of the tree is in the branches and leaves above the ground. Then the moments of inertia of the tree before and after felling are \begin{align} I_\text{tree,up} &= m \left( (R+h)\cos\theta \right)^2 \\ I_\text{...

4

Cutting the trees and leaving them flat instead of vertical will diminish the moment of inertia of earth. The angular momentum of course will not change, but the speed of rotation will increase. However, I do not believe that the change is measurable with current instruments.

4

I know the example given sounds crazy but the physics behind it might be useful for someone learning rotational dynamics. The angular momentum of a system does not change if there are no external torques, since $$\frac{d\vec L}{dt}=\vec \tau,$$ where $\vec L$ is the total angular momentum and $\vec \tau$ is the total external torque. So if you cut the trees ...

1

To formalize @knzhou's comment: The answer, in a nutshell, is no. Your basic assumption that cutting trees reduces the Earth's mass is wrong, because trees don't leave the Earth when they are cut! Even if all trees left the Earth when cut, a lot of tree cutters plant trees to replace what they cut, and each tree is such a tiny amount of mass ...

0

Apart from friction in the load (which cannot be predicted), the flywheel does not make any difference to the maximum rotation speed reached by the motor. If there is no friction, then given sufficient time the motor can accelerate the flywheel up to its own maximum unloaded speed, however small the torque supplied by the motor. An even higher speed can be ...

2

The angular momentum is $\vec L~=~\vec r\times\vec p$ which according to a bulk system with a moment of inertial $I$ is also $\vec L~=~\hat n I\omega$. Here the unit vector $\hat n$ is normal to the plane of $\vec r$ and $\vec p$. In Hamiltonian mechanics we have $${\dot L}_i~=~I\dot\omega_i~=~\{H,~L_i\}_{pb}~=~0,$$ for $i$ the coordinate direction which ...

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