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The "associated scalar equation" is just the formula for the time evolution of the scalar magnitude of the displacement, $r$, rather than all its vector components. It really only makes sense to write such an equation if the right-hand side can be expressed in terms of $r$ only, and not $\mathbf{r}$. Then you can use it to analyze the evolution of $r$ in ...

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Hint. I think the error is in your first equation, adding the forces for mass M1: $\vec{F}_{M_3}+\vec{F}=m_{1}\vec{a}_1\\$. There is an additional force on M1 that you have omitted. Edit: The pulley exerts a force on M1.

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You are introducing some irrelevant variables, as $F_{M_3}$, $T_1$, $T_2$. Let us make the assumption that $T_1=T_2=T$ (the pulley doesn't rotate and the string is massless). The whole has mass $M=m_1+m_2+m_3$, accelerates with $a$ and the force on $M$ is $$F=Ma.$$ The tension $T$ equals $m_2 a$ and also $m_3g$, so$$a=\frac{m_3}{m_2}g.$$Hence ...

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Some conservation laws are related to conservation of angular momentum. There is a famous example (from Feynman if I recall correctly), where you assume an infinite flat space and conservation of angular momentum about any point, and then you get conservation of linear momentum for free. Intuitively, to get say the $x$ component of linear momentum is ...

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No. As @Jim said, the heat would weaken the rock, which would cause a tunnel collapse before any sublimation could occur. Also, remember that the air in the tunnel would generally be at the same temperature as the rock (unless a large cooling system was put in), so thermal equilibrium would be maintained without any sublimation.

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when low mass object hits high mass object it is reflected gaining opposite velocity almost the same as initial velocity. If I jump onto the wall why my body is not reflected? I know that collision is not fully elastic but it should be at least similar. Human body is not elastic: it cannot be deformed/ compressed in any way and then return to ...

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The angular momentum $L_{A/B}$ of a rigid body $A/B$ about its center of mass is $$L_{A/B} = I_{A/B} \omega_{A/B},$$ where $I_{A/B}$ is the inertia matrix of $A/B$ about its center of mass in the world frame and $\omega_{A/B}$ is the angular velocity of $A/B$. The angular momentum $L_{A/B}^0$ of a rigid body $A/B$ about the origin of the world frame is ...

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The rocket is in free fall along with the book. The nearest gravitating bodies are very far away, so whatever meager acceleration they cause will be almost exactly the same on the rocket and on the book. Suppose you instead went on a very close flyby of a neutron star. Now the book will fall rapidly, and away from the rocket's center of mass. The only way ...

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Thermally supported, self-gravitating bodies (those to which the Virial theorem applies) qualify if you are willing to neglect the radiative energy loss. Depending on the time-scales that interest you this can be quite a good approximation. Stellar nebulae, brown dwarfs and so on.

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