Mass–energy equivalence - Wikipedia
Motion is the action of changing location or position. The general study of the relationships between motion, forces, and energy is called mechanics. With power and energy, power is units of energy divided by time. The same difference as distance and velocity. The units of power are watts. Energy is an abstract scalar quantity associated with motion (kinetic energy) or arrangement (potential energy). roller coaster; waterwheel; hydroelectric power .
To be specific, power is defined as the work done divided by the time that it took to do that work.
Work and energy
We already said that both weightlifters are doing 1, joules of work. The weightlifter on the right takes 1 second to lift his weights, and the weightlifter on the left takes 3 seconds to lift his weights. If we plug those times into the definition of power, we'll find that the power output of the weightlifter on the right during his lift is 1, joules per second. And the power output of the weightlifter on the left during his lift is joules per second.
A joule per second is named a watt, after the Scottish engineer James Watt. And the watt is abbreviated with a capital W. All right, let's look at another example. Let's say a 1, kilogram car starts from rest and takes 2 seconds to reach a speed of 5 meters per second. We can find the power output by the engine by taking the work done on the car divided by the time it took to do that work.
To find the work done on the car, we just need to figure out how much energy was given to the car. In this case, the car was given kinetic energy and it took two seconds to give it that kinetic energy. If we plug in the values for the mass and the speed, we find the engine had a power output of 6, watts. We should be clear that what we've really been finding here is the average power output because we've been looking at the total work done over a given time interval.
If we were to look at the time intervals that got smaller and smaller, we'd be getting closer and closer to the power output at a given moment. And if we were to make our time interval infinitesimally small, we'd be finding the power output at that particular point in time. We call this the instantaneous power. Dealing with infinitesimals typically requires the use of calculus, but there are ways of finding the instantaneous power without having to use calculus.
For instance, let's say you were looking at a car whose instantaneous power output was 6, watts at every given moment. Since the instantaneous power never changes, the average power just equals the instantaneous power, which equals 6, watts.
In other words, the average power over any time interval is going to equal the instantaneous power at any moment. And that means work per time gives you both the average power and the instantaneous power in this case.
Let's say you weren't so lucky, and the instantaneous power was changing as the car progressed. Then, how would you find the instantaneous power? Well, we know that power is just the work per time. So something we can try is to plug in the formula for work, which looks like FD cosine theta, and then divide by the time. Something that you might notice is that now we have distance per time in this formula. This circumstance has encouraged the false idea of conversion of mass to energy, rather than the correct idea that the binding energy of such systems is relatively large, and exhibits a measurable mass, which is removed when the binding energy is removed.
The difference between the rest mass of a bound system and of the unbound parts is the binding energy of the system, if this energy has been removed after binding.
For example, a water molecule weighs a little less than two free hydrogen atoms and an oxygen atom. The minuscule mass difference is the energy needed to split the molecule into three individual atoms divided by c2which was given off as heat when the molecule formed this heat had mass.
Likewise, a stick of dynamite in theory weighs a little bit more than the fragments after the explosion, but this is true only so long as the fragments are cooled and the heat removed. Such a change in mass may only happen when the system is open, and the energy and mass escapes. Thus, if a stick of dynamite is blown up in a hermetically sealed chamber, the mass of the chamber and fragments, the heat, sound, and light would still be equal to the original mass of the chamber and dynamite.
If sitting on a scale, the weight and mass would not change. This would in theory also happen even with a nuclear bomb, if it could be kept in an ideal box of infinite strength, which did not rupture or pass radiation. If then, however, a transparent window passing only electromagnetic radiation were opened in such an ideal box after the explosion, and a beam of X-rays and other lower-energy light allowed to escape the box, it would eventually be found to weigh one gram less than it had before the explosion.
This weight loss and mass loss would happen as the box was cooled by this process, to room temperature. However, any surrounding mass that absorbed the X-rays and other "heat" would gain this gram of mass from the resulting heating, so the mass "loss" would represent merely its relocation.
Thus, no mass or, in the case of a nuclear bomb, no matter would be "converted" to energy in such a process. Mass and energy, as always, would both be separately conserved. Massless particles[ edit ] Massless particles have zero rest mass.
This frequency and thus the relativistic energy are frame-dependent. If an observer runs away from a photon in the direction the photon travels from a source, and it catches up with the observer—when the photon catches up, the observer sees it as having less energy than it had at the source.
The faster the observer is traveling with regard to the source when the photon catches up, the less energy the photon has. As an observer approaches the speed of light with regard to the source, the photon looks redder and redder, by relativistic Doppler effect the Doppler shift is the relativistic formulaand the energy of a very long-wavelength photon approaches zero.
This is because the photon is massless—the rest mass of a photon is zero. Massless particles contribute rest mass and invariant mass to systems[ edit ] Two photons moving in different directions cannot both be made to have arbitrarily small total energy by changing frames, or by moving toward or away from them.class 11 physics chapter 6 - Work, Energy and Power 01 - Introduction - Formulae for Work IIT JEE
The reason is that in a two-photon system, the energy of one photon is decreased by chasing after it, but the energy of the other increases with the same shift in observer motion. Two photons not moving in the same direction comprise an inertial frame where the combined energy is smallest, but not zero. This is called the center of mass frame or the center of momentum frame; these terms are almost synonyms the center of mass frame is the special case of a center of momentum frame where the center of mass is put at the origin.
The most that chasing a pair of photons can accomplish to decrease their energy is to put the observer in a frame where the photons have equal energy and are moving directly away from each other. In this frame, the observer is now moving in the same direction and speed as the center of mass of the two photons.
The total momentum of the photons is now zero, since their momenta are equal and opposite. In this frame the two photons, as a system, have a mass equal to their total energy divided by c2.
This mass is called the invariant mass of the pair of photons together. It is the smallest mass and energy the system may be seen to have, by any observer. It is only the invariant mass of a two-photon system that can be used to make a single particle with the same rest mass.
Energy, power and action - Physics Stack Exchange
If the photons are formed by the collision of a particle and an antiparticle, the invariant mass is the same as the total energy of the particle and antiparticle their rest energy plus the kinetic energyin the center of mass frame, where they automatically move in equal and opposite directions since they have equal momentum in this frame. If the photons are formed by the disintegration of a single particle with a well-defined rest mass, like the neutral pionthe invariant mass of the photons is equal to rest mass of the pion.
In this case, the center of mass frame for the pion is just the frame where the pion is at rest, and the center of mass does not change after it disintegrates into two photons. After the two photons are formed, their center of mass is still moving the same way the pion did, and their total energy in this frame adds up to the mass energy of the pion. Thus, by calculating the invariant mass of pairs of photons in a particle detector, pairs can be identified that were probably produced by pion disintegration.
A similar calculation illustrates that the invariant mass of systems is conserved, even when massive particles particles with rest mass within the system are converted to massless particles such as photons. In such cases, the photons contribute invariant mass to the system, even though they individually have no invariant mass or rest mass. Thus, an electron and positron each of which has rest mass may undergo annihilation with each other to produce two photons, each of which is massless has no rest mass.
However, in such circumstances, no system mass is lost. Instead, the system of both photons moving away from each other has an invariant mass, which acts like a rest mass for any system in which the photons are trapped, or that can be weighed.
Thus, not only the quantity of relativistic mass, but also the quantity of invariant mass does not change in transformations between "matter" electrons and positrons and energy photons. Relation to gravity[ edit ] In physics, there are two distinct concepts of mass: The gravitational mass is the quantity that determines the strength of the gravitational field generated by an object, as well as the gravitational force acting on the object when it is immersed in a gravitational field produced by other bodies.
The inertial mass, on the other hand, quantifies how much an object accelerates if a given force is applied to it. The mass—energy equivalence in special relativity refers to the inertial mass. However, already in the context of Newton gravity, the Weak Equivalence Principle is postulated: Thus, the mass—energy equivalence, combined with the Weak Equivalence Principle, results in the prediction that all forms of energy contribute to the gravitational field generated by an object.
This observation is one of the pillars of the general theory of relativity.