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5 votes
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Time differential between two signals sent at two different near-relativistic speeds

This is a very interesting question. Indeed, as a one-way-speed-of-light (OWSOL) experiment the answer is a duplicate of all of the other myriad OWSOL questions: there is no possible experiment which ...
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Why do equation of motion fail to apply in non-inertial frame even after applying pseudo force?

Your computation in the inertial frame seems to be correct. However, car 2's deceleration in car 1's frame is wrong. Let's look at this in detail. Let $D$ be the distance between the car when they ...
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2 votes
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What does size of an inertial frame mean?

The Equivalence Principle (EP) is valid locally: if you perform a local experiment in an inertial frame of reference, you can't say if you're freefalling or not falling, you can't say if you're ...
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2 votes

What does size of an inertial frame mean?

I've never seen it phrased like that, but I understand it as the following. Strictly speaking, the frame is defined locally, on a point. So the further you move from that point, the less inertial the ...
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1 vote

What does size of an inertial frame mean?

I don't have the book but I think he means the following. At any point in spacetime you can assign an inertial frame, similar to how you can assign a tangent line to a point on a curve. The size of ...
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0 votes

The chronology of taking approximation changes the answer?

I've found out where the error was, it was not due to chronology of taking approximations, instead I had done error while approximating $\arctan(\sqrt{\frac{h_0}{R}})$. Let $x=\frac{h_0}{R}<<1$, ...
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-2 votes

Time differential between two signals sent at two different near-relativistic speeds

Photons are bound by relativity. Even if a really fast vessel. It is o ly when an object approaches 1/10th c that dilation becomes heavily noticeable. One aspect of Special Relativity speed of light ...
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1 vote
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How can one calculate the torque around this point?

The short answer What you did wrong, is you forgot to the contribution of momentum to angular momentum. $$ \vec{L}_P = \vec{L}_{\rm COM} + \vec{r}_{\rm COM} \times \vec{p} $$ The second thing wrong is ...
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1 vote

For someone staying on Earth, what is the minimum possible time to send a spaceship to Alpha Centauri and have it back?

The minimum time, as measured on Earth, for anything to travel to Alpha Centauri and back is 8.74 years or thereabouts, if the something was massless. For something massive like a spaceship the time ...
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2 votes
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For someone staying on Earth, what is the minimum possible time to send a spaceship to Alpha Centauri and have it back?

As the spaceship accelarates, at which point the time dilation added due to the incrase of speed starts to increase the total time of the journey measured on Earth? Never. If the ship can accelerate,...
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2 votes
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Are curvilinear coordinates inertial?

Indeed, it is valid to consider that polar coordinates are non-inertial. You should be aware that the term “reference frame” does not have one unique meaning. Different authors may use it to mean ...
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Are curvilinear coordinates inertial?

Reference frames can be inertial or non inertial. Coodinate systems are not reference frames unless the frame is somehow being tied to the coordinate system. Does the book explain how the frame is ...
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Why the proper time of photon is zero?

Because for a massless particle $ds^2=0$.
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9 votes
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How could any frame of reference be inertial?

In newtonian mechanics, inertial frames are an equivalence class. They can be defined as frames where Newton's laws are valid. If you can find one inertial frame, then you automatically get an ...
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A question about relativity and time dilation from a lay-person

The best way to see what's going on is with a special Minkowski Diagram called a Lodel Diagram. Let's say the unprimed frame is on Earth, and the primed frame is the returning rocket ship. Since you ...
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A question about relativity and time dilation from a lay-person

Time dilation can be confusing until you view it in the right way, and then it becomes obvious. Time dilation is caused because clocks in two moving frames are out of synch relative to each other- it ...
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A question about relativity and time dilation from a lay-person

Alice is on earth. Bob travels from Far Away. The story according to Alice: At 2PM, Bob started his journey, with his clock correctly set to 2PM (as was mine). The journey took him four hours, but ...
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A question about relativity and time dilation from a lay-person

When you say "he will reach the Earth in one week" you have to also specify whether this is as measured by A or as measured by B. They have different coordinate systems. They're both ...
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A question about relativity and time dilation from a lay-person

It is better to think of a row of ships travelling to earth with the same velocity of A (Alice) and with synchronized clocks. And a row of stationary clocks synchronized with the B (Bob) earth's clock....
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1 vote
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Calculation of center of mass of a bar with linear density by branches

yes. remember that the CoM of a continuous mass distribution $\mathbb M$ with mass density $\mathrm dm = \rho(\mathbf{\vec r}) \, \mathrm dV$ is defined as $$\mathbf{\vec R} = \dfrac1M \int_\mathbb ...
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Is there tested proof of time dilation in the stationary frame, from the perspective of the moving frame?

Directly testing that each inertially moving clock observes that the other is moving slower is difficult, mainly because fast moving inertial objects will typically leave the vicinity of the Earth. So ...
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Null conserved angular momentum

First of all, the angular momentum is defined only with respect to a point. If we treat that point as the origin, and the position of the particle is $P(t)$, then angular momentum is $mP(t)\times P'(t)...
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Observing from a reference frame moving with high speed

Both your approach and the book's approach are fine. There's no absolute motion, only relative motion, so we can either take the Earth to be at rest, or the rocket to be at rest. Your approach takes ...
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1 vote

Null conserved angular momentum

The angular momentum is defined relative to a chosen point. For it to be zero, the velocity vector of the center of mass of a non-rotating rigid object must be directed toward that point.
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8 votes
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Null conserved angular momentum

Yes, if the angular momentum of a particle is conserved and it is zero, the particle must move along a straight line. Indeed, by using the triple product identity, we have $$ {\bf r} \times ( {\bf r} \...
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Would an accelerated-time space region deflect projectiles?

Time runs slower in a gravitational well than far above. From this, Einstein derived that light is deflected when it passes near the Sun. He predicted that a star whose apparent position is very near ...
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4 votes
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Dependence (or lack thereof) of forces on frames of reference

When you say "blocks A and B move with a relative acceleration of -3 m/s2" you are considering the motion of one block in the reference frame attached to the other block (block C). But the ...
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1 vote
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Galilean Transformations Derivation

Let's analyze the spacetime coordinate from the Greek perspective. The Greek will see that the Roman will approach him/her an meet at origin coordinate. Then the Greek perform some measurement at ...
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2 votes
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Find the equation for the angle $\theta$ in which the particle leaves the semicircle. No Friction

I don't think this is wrong. $k=0$ represents the case where the large hemisphere has zero mass, so no inertia. I think you are looking for the case where the large hemisphere is "fixed", so ...
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How does attaching a point mass to the CM of an object affect its moment of inertia about the CM?

I may be wrong, but if you're referring to these equations when you say "formulas," then your intuition is right that the moment of inertia doesn't change, despite the "total mass" ...
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How does attaching a point mass to the CM of an object affect its moment of inertia about the CM?

We can test a simple case. Two point masses $M$ separated by a fixed distance $2R$. If the object is rotating, the angular momentum from the COM is $L = RMv + RMv = 2MR\omega R = 2MR^2 \omega$. So $I =...
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How does attaching a point mass to the CM of an object affect its moment of inertia about the CM?

For a point mass $I=mr^2$, where $r$ is the distance from the centre of rotation, so it is zero. The total mass in other formulae comes from the shape the formula assumes - most formulae for shapes ...
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How and where does air that is displaced by a constant moving vehicle flow?

Consider a hallway with people walking at constant speed in one direction. Now imagine there is a partial obstruction, so the hallway becomes narrower. People will move faster there, so the same ...
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1 vote

How do we assume the direction of $u_{\theta}$ and $u_{r}$ in polar coordinate systems?

As a rule, $\hat{u}_X$ is always pointing in the direction along which $X$ grows. It works when $X$ is a linear parameter ($x$, $z$, $r$...) as well as when it's an angular parameter ($\theta$, $\phi$....
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1 vote

How do we assume the direction of $u_{\theta}$ and $u_{r}$ in polar coordinate systems?

The unit radial vector is always away from the origin. The unit tangential vector is always anti-clockwise around the origin such that $\hat{r}\times\hat{\theta}$ is out of the diagram. Not knowing ...
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10 votes

Wouldn't the cosmic background radiation (CMB) produce drag and thus create a preferential inertial frame?

The other answers are already good, but I'll take a stab at an extremely short answer, that hopefully cuts at the heart of your confusion: Yes, the CMB creates a preferred frame, but so does any other ...
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0 votes

Cart Pole kinetic energy

Let us start with an example first. Consider a pendulum with constantly accelerated support (instead of the support being a degree of freedom of the system). The position of said pendulum can be ...
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3 votes

Wouldn't the cosmic background radiation (CMB) produce drag and thus create a preferential inertial frame?

There's a distinction between phenomena and physical laws. Phenomena are what actually exists in the universe. Physical laws are what would happen given particular circumstances. Physical laws are the ...
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2 votes

Wouldn't the cosmic background radiation (CMB) produce drag and thus create a preferential inertial frame?

One thing to note is that the CMB looks different from different locations (and at different times). So while it provides a local frame of reference for every point in space-time, that frame is not ...
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2 votes
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Why can't we use integral of $x$, $y$ and $z$ in calculating moment of inertia

The moment of inertia is defined for a specific rotation axis. The radius $r$ is the distance from this axis, not from the origin point. For example, for the moment of inertia around the $z$-axis the ...
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12 votes

Wouldn't the cosmic background radiation (CMB) produce drag and thus create a preferential inertial frame?

Indeed the existence of the CMB constitutes a reference of velocity. The thing is: the CMB is not unique in that regard. As a thought experiment: what is still possible in a Universe that is ...
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1 vote

Is this a rigorous derivation of Coriolis effect? It seems pretty simple

In my earlier answer I derived the coriolis effect for the general case. The general case is that the circumnavigating object can have any direction of velocity with respect to the rotating coordinate ...
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28 votes

Wouldn't the cosmic background radiation (CMB) produce drag and thus create a preferential inertial frame?

The CMB does in fact produce a preferential reference frame. Even without pressure, the preferential frame would be the one that equalizes the red and blue shift in all directions. For example Earth'...
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2 votes

Is this a rigorous derivation of Coriolis effect? It seems pretty simple

Let me first give a general derivation for the case of an object that is physically going around a center of attraction, with the provided centripetal force dialed in to be equal in magnitude and ...
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1 vote
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Is this a rigorous derivation of Coriolis effect? It seems pretty simple

indeed you can use it for three dimensional case start with the angular momentum $$\vec L=\mathbf I\,\vec\omega$$ with the inertia tensor for a mass point $$\mathbf I=-m\,\mathbf{r}_\times\,\mathbf{r}...
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2 votes

Euler's Angles and Uniquely Defining the Orientation of a Rigid Body

Euler angles do not solve this problem 100%. There are cases where two or more sets of angles result in the same final orientation. We call these the degenerate cases, or gimbal lock, or singularities....
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14 votes
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How is it that we are still able to talk about the speeds of other galaxies, distances and times across the universe, etc despite GR?

There are two issues mixed up in your question. First, there is a preferred cosmic rest frame: the frame of the cosmic microwave background. This does not imply any contradiction with the principle of ...
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1 vote
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How do the inertia tensor varies when a rigid body rotates in space?

This has been answered below, but consider the rotation matrix $\mathrm{R}$ whose columns represent the local $\hat{x}$, $\hat{y}$ and $\hat{z}$ axis in the world coordinates. $$ \mathrm{R} = \begin{...
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4 votes
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When two systems of forces acting on a rigid body are equivalent?

The equations of motion can be thought of given a specified motion, provided by the translation acceleration vector of the center of mass $\boldsymbol{a}_C$ and the rotation acceleration vector of the ...
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3 votes

When two systems of forces acting on a rigid body are equivalent?

A theorem for a rigid body is: "Every system of forces is equivalent to a single force through an arbitrary point, plus a couple (either or both of which may be zero.)". A couple is defined ...
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