newtonian mechanics - Rigid bodies - the wheel

As I've been taught lately in my mechanics course:

the wheel has a unique property: at every moment of motion, the touching point between the wheel and the ground is not in movement and therefore no work is done by the friction force.

Now, many of those problems are solved by using the 2nd Newton law and its rotational analog.

For instance consider having a wheel with a mass $m$ and a radius $R$ rolling on a slope that creates an angle of $\theta$ and we want to calculate its acceleration then we can start by writing: $$ma=mg\sinθ−F_f$$

and the analog equation for torque: $$F_f R=I\alpha.$$

where $F_f$ is the frictional force. Now, the first equation is the 2nd Newton law applied on the centre of mass of the wheel, and as we see, one of the forces is the external frictional force. Now, though the touching point is not in movement at the moment, the center of mass is, and in the equation we assume there is a friction force on the center of mass and therefore work is done indeed. Now, after thinking about this for a while, I've come to the conclusion that this makes sense, cause if we see the wheel as point of mass located in the center, then energy is not preserved because some of it is transfered to the spin and that's why we have the second equation.

The question I'm having trouble with is whether the "work" of the friction force on the center of mass is equal to the energy transfered to the spin of the wheel?

Answer

The best treatment that I've seen of these types of questions comes from Sherwood and Chabay, in Matter and Interactions.

If you look at the wheel as a particle (the "point particle" system), then it cannot rotate, because particles have no physical extent. That means that the distance in the definition of work is the distance that the center of mass travels. That also means that the particle-wheel can only have translational kinetic energy. Let the displacement of the center of mass be $\Delta x$, which is a distance $d$ down the plane.

$$W_{net,PP} = m\vec{g}\cdot\Delta x+\vec{F}_s\cdot\Delta x = mgd-F_sd= \Delta E = \frac{1}{2}mv_f^2$$

If, however, the wheel is modeled as a physical object (the "real" system), then the point of application of each force is its real contact point, which isn't moving for the friction force, but is moving for the weight (because it's the CM). However, it can now have rotational kinetic energy.

$$W_{net,R} = m\vec{g}\cdot\Delta x+{F}_s\cdot0 = mgd = \Delta E = \frac{1}{2}mv_f^2 + \frac{1}{2}I\omega_f^2$$

Combining the expressions shows that:

$$F_sd = \frac{1}{2}I\omega_f^2$$

You could also modify the system in either case to include the Earth in the system, which would convert that positive work done by gravity on the LHS to a loss in $U_g$ on the RHS.

electromagnetism - Induced magnetic field produces electric field and vice versa forever!

So here are the two of Maxwell's laws that I am interested in:

So we have the simple circuit (from google):

So, before the system goes into steady-state we know that charge slowly accumulates on the plates of the conductor. So the charge on the plates gets bigger and bigger while the charges that carry the current get smaller and smaller, so the current gets weaker.
Applying Ampere's law on the wire we find the induced magnetic field due to the current $I(t)$ that penetrates the surface $\Sigma$ (see the integral of $\mathbf{J}\mathrm{d}\mathbf{S}$) and not due to an electric field.
Now, this induced magnetic field is changing with respect to time (because current is changing). But from the Maxwell-Faraday equation we conclude that this changing magnetic field will produce an electric field which again changes with respect to time. And then we have another induced magnetic field due to that changing electric field. And the cycle goes on.
So, am I right? And if I am, when does this stop? And how does it changes the way I calculate each induced field? Does it have to do with electromagnetic waves?

homework and exercises - Treating gravitational lensing as index of refraction

In Einstein's theory of gravity, an electromagnetic wave passing near a massive object is bent from its rectilinear path. We may regard this bending equivalently as due to a medium of refractive index $$\mu_g = 1 - \frac{2\phi}{c^2}$$
where $\phi$ is the gravitational potential due to an object of mass $M$.

Consider radiation for a Quasar Q (treated as point like object) reaching observer P as shown in figure. Here $b$ is the distance of closes approach to gravitational object $M$. Show that the optical path length for path QP is given by $$d = d_1 + d_2 + \frac{2MG}{c^2}\left(\ln\frac{4 d_1 d_2}{b^2}\right)$$

• What does "we may regard this bending equivalently as due to a medium of refractive index" mean? If we can treat it as a medium, them what do we take the angle of incidence, refraction, the medium interface etc to be?


• By optical path length of path QP, should we find $\int n(s)ds$ over the line segment QP or should we do it over the actual path taken by light to move from Q to P? And how do we show the last part?

Answer

ANSWER TO :

DOUBT 1: They Are basically modelling the curving light ray to be the same scenario as when a light beam passes through a medium having continuously varying R.I for each infinitesimal layer(like the atmosphere suppose). The function of R.I here is given as a function of the potential, which is a function of the distance from the object M.

DOUBT 2: You have to find the optical path length $\Delta$, which by definition is:

$\Delta = \int_{C} n(s).ds$ where C denotes the curve signifying its actual path traversed.

electricity - Why the bulb does not glow in this configuration of batteries?

I connected a bulb to a battery positive terminals with positive and negative terminals with negative . It glows as it should but when i connect the positive terminal of the same bulb to the positive terminal of one battery and negative terminal of the same bulb to the negative terminal of another battery. The bulb does not glow.

What I want to ask here is that electric potential difference is being maintained then why does the bulb does not glow. NOTE : Batteries are not connected with each other. Both batteries are separately placed.

Why is quantum entanglement considered to be an active link between particles?

From everything I've read about quantum mechanics and quantum entanglement phenomena, it's not obvious to me why quantum entanglement is considered to be an active link. That is, it's stated every time that measurement of one particle affects the other.

In my head, there is a less magic explanation: the entangling measurement affects both particles in a way which makes their states identical, though unknown. In this case measuring one particle will reveal information about state of the other, but without a magical instant modification of remote entangled particle.

Obviously, I'm not the only one who had this idea. What are the problems associated with this view, and why is the magic view preferred?

Answer

Entanglement is being presented as an "active link" only because most people - including authors of popular (and sometimes even unpopular, using the very words of Sidney Coleman) books and articles - don't understand quantum mechanics. And they don't understand quantum mechanics because they don't want to believe that it is fundamentally correct: they always want to imagine that there is some classical physics beneath all the observations. But there's none.

You are absolutely correct that there is nothing active about the connection between the entangled particles. Entanglement is just a correlation - one that can potentially affect all combinations of quantities (that are expressed as operators, so the room for the size and types of correlations is greater than in classical physics). In all cases in the real world, however, the correlation between the particles originated from their common origin - some proximity that existed in the past.

People often say that there is something "active" because they imagine that there exists a real process known as the "collapse of the wave function". The measurement of one particle in the pair "causes" the wave function to collapse, which "actively" influences the other particle, too. The first observer who measures the first particle manages to "collapse" the other particle, too.

This picture is, of course, flawed. The wave function is not a real wave. It is just a collection of numbers whose only ability is to predict the probability of a phenomenon that may happen at some point in the future. The wave function remembers all the correlations - because for every combination of measurements of the entangled particles, quantum mechanics predicts some probability. But all these probabilities exist a moment before the measurement, too. When things are measured, one of the outcomes is just realized. To simplify our reasoning, we may forget about the possibilities that will no longer happen because we already know what happened with the first particle. But this step, in which the original overall probabilities for the second particle were replaced by the conditional probabilities that take the known outcome involving the first particle into account, is just a change of our knowledge - not a remote influence of one particle on the other. No information may ever be answered faster than light using entangled particles. Quantum field theory makes it easy to prove that the information cannot spread over spacelike separations - faster than light. An important fact in this reasoning is that the results of the correlated measurements are still random - we can't force the other particle to be measured "up" or "down" (and transmit information in this way) because we don't have this control even over our own particle (not even in principle: there are no hidden variables, the outcome is genuinely random according to the QM-predicted probabilities).

I recommend late Sidney Coleman's excellent lecture Quantum Mechanics In Your Face who discussed this and other conceptual issues of quantum mechanics and the question why people keep on saying silly things about it:

http://motls.blogspot.com/2010/11/sidney-coleman-quantum-mechanics-in.html

fluid dynamics - Bubble in a pipeline

I am just thinking about this phenomenon: We have a horizontal pipeline with a flowing liquid, which contains a small bubble of gas. How do the dimensions of this bubble change when it reaches a narrower point of the pipeline? Are there practical applications that track bubble sizes to estimate the properties of the flow?

general relativity - The Fabric of Space-time?

I am not an academic in anyway, just someone interested in the story that is our universe. So my apologies if this isn't a well thought out inquiry.

I've been struggling with a concept for some time (which is probably rudimentary) that I can't even frame into the right words to know where to begin a Google search, so thought I'd try this platform.

I'm trying to wrap my mind around the fabric of space. I can understand the effect of a mass like our Sun and its gravitational influence on the fabric of space, and how the planets fall into indents orbiting that mass. But my questions is, are all the masses in the universe on top of that "fabric"? Essentially, I'm asking if the fabric of space is all encompassing in the universe, is it at all points North, East, South, West, above, below, in between, or just a horizontal plain that all masses reside upon? I know that space-time twists in relation to a mass, is space-time manipulated the same way above the Earth and Sun as it is below?

Not sure if I'm framing this question correctly, but any insight you can provide would be beneficial.