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1

Both in classical as in quantum mechanics, energy is the conserved charge (in the sense of the Noether's theorem) of a system (Hamiltonian) which is invariant up to time translations. This is the most general definition of energy. It does not depend on the nature of the system (being either matter or radiation). Also note that at least in special ...


0

The curvature of spacetime is determined by the stress-energy tensor, and in the stress-energy tensor we do not distinguish between matter and energy. The two are treated as equivalent and interconverted using Einstein's famous equation $E = mc^2$. The Schwarzschild radius of a mass is conventionally written as: $$ r_s = \frac{2Gm}{c^2} \tag{1} $$ where ...


0

Does a ticking watch have more mass? Yep. As does a wound-up watch, or a hot watch, or a watch that's higher up in a gravitational field. It claims that a ticking watch has more mass then a non ticking watch due to the intrinsic KE, PE and thermal energy of the watch's internal movements that manifests itself as part of the watch's mass. Is this ...


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What's wrong with it is that they didn't define the two watches very clearly. It's obvious to assume they meant identical watches (my watch is more massive than my wife's, ticking or not) but they don't specify the conditions of the watches. Really though, we only need one watch and an arbitrarily accurate scale to show what's going on. Let's say our watch ...


0

It is correct to say that we measure the mass of moving objects to be greater than an object at rest relative to our reference frame. Another way of saying moving object is- an object with kinetic energy. Think about it this way, if moving objects didn't appear to gain mass then as you give them more kinetic energy they could accelerate to the speed of ...


1

It's not so much that it is ticking that is crucial here but the fact that the watch is in a higher energy state that in its, say unwound state. This fact increases the rest mass of the watch by an amount $\Delta E/c^2$, where $\Delta E$ is the potential energy input to elastically stress the spring and thus wind the watch. A wound, broken, unticking watch ...


2

We can write total energy $E$ two ways: \begin{equation} E^2=p^2c^2+m^2c^4 \\ E=T+mc^2, \end{equation} where $T$ is kinetic energy. Eliminating $E$ from those two equations will give you the desired result.


2

Hint: $T = E - E_0 = m\gamma c^2 - mc^2 = mc^2(\gamma -1)$ and $p = |\vec p| = m\gamma |\vec v| = m\gamma v$


2

Yes, bonds have mass, like every other kind of energy. This can be significant; if you had a glueball (a hypothetical particle made of massless gluons), it would have mass, and all of the mass would be from the bond energy! Same would go if you somehow managed to bind photons together.


20

As far as the theory goes, you are absolutely correct, the (negative) binding energy between atoms in a molecule contributes to the total mass of that molecule, so a stable molecule is less massive than the sum of the masses of its constituent atoms. However (as you yourself calculated), the mass difference is absolutely tiny, and as far as I know, it has ...


1

The equation $E = m^2c^2 + p^2 c^2$ is restricted to Special Relativity. However, in classical physics we have $$ \vec{F} = m \vec{a}, $$ and $$\vec{F} = m \vec{g},$$ whence $$ m \vec{a} = m \vec{g}. $$ This can be written as $$ m \big( \vec{a} - \vec{g}\big) = \vec{0}. $$ From a mathematical point of view we have $$ \big( m = 0 \big) \vee \big( \vec{a} - ...


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As in Ben Crowell's Answer, the concept of "Relativistic Mass" is not wrong, but it is awkward. There are several things a loose usage of the word "mass" could imply, all different and thus it becomes a strong convention to talk about the meaning of the word "mass" that is Lorentz invariant - namely the rest mass, which is the square Minkowski "norm" of the ...


-1

Look at the paragraph "gravity and the photon" in this link: In the relativistic framework, i.e. large velocities, any energy is also a relativistic mass: For the photon this means the following equation: m is the relativistic mass of a photon with energy h*nu. Gravity attracts relativistic mass, and the photon has one. Read the link further to ...


1

You misunderstand. All objects have some escape velocity, which is the velocity needed for anything (photons or matter) to escape from that object's gravitational field. And that's not the velocity it needs to maintain under some sort of constant thrust, but the initial velocity it needs to, shall we say, coast away from the object. For a black hole that ...


0

One cannot obtain "clean" energy which is completely free off momentum, and cannot obtain "clean" matter which is free off momentum and potential energy. So question is ill-posed, there is no "clean" states which can be described as "energy into matter". That just cannot happen. When we consider reactions of elementary particles, the most common scenario ...


0

In what ways can energy transform into matter and vice versa? I am sure in special relativistic theory there is no such transformation. Why? Energy is an abstract mathematical quantity obeying local conservation law. Matter is a basic thing the world is made of. It is not a mathematical concept. One can quantify one aspect of it, say introduce ...


4

The force between nucleons is rather complicated, so let's consider the simpler example of assembling a hydrogen atom from a proton and an electron. We start with the proton and electron at rest and a long way apart. The kinetic energy is zero (because they're at rest) and the potential energy is given by: $$ V = -k\frac{Q_1Q_2}{r^2} \tag{1} $$ where ...


0

People are mistaking mass for force. An object moving does not have more mass, it has inertia, and moving it away from the earth weakens the gravitational attraction between the earth and the object, causing a weight change. However, weight is a measurement of gravitational force on between two bodies containing mass. Mass, however, cannot and does not ...


2

Since you have not specified the "real world" size of the ship, let's take a 74-gun ship of the line https://en.wikipedia.org/wiki/Seventy-four_(ship) as the desired type, firing a 36-pound cannon. The bore on such a cannon https://en.wikipedia.org/wiki/36-pounder_long_gun was about 175 mm, with a shot weight of about 39 lb. Shrinking this cannon to a bore ...


1

Short: No Because: Assume material SG ~= 7 Assume ~= non-sonic shot. Say 500 m/S based on this superb reference R = 0.05mm = 5E-5m E= 1/2 mV^2 m kg = 4/3.Pi.R^3 x sg x 1000 kg/m^3 = 4/3x 3.14 x (5E-5)^3 x 1000 x 7 = 3.E-9 kg At say V = 500 m/s E = 0.5 x 3.5E-9 x 500^2 ~~= 500 micro Joule. (438 uJ calc) A VERY solid teaspoon weighs 50 gram. ...



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