The post Special relativity: length contraction and time dilation appeared first on Physics Capsule.

]]>First of all, Irene, thank you for the great question. To explain exactly how the characters in *Interstellar* stay the same age we would have to either introduce a considerable amount of mathematics or hope that you would to be satisfied with vague descriptions. Since neither of those sound interesting we will have to seek an indirect route.

We can begin with the special theory of relativity, which is simpler and much easier to understand, and use that to examine how time can actually run at different rates. Once we are convinced of this flexibility of time, a verbal explanation of gravitational time dilation (which is really what you are looking for in your question) will be much more convincing.

We talked about inertial frames of reference while discussing Newton’s first law and we will use the same idea here to describe what are called Lorentz transformations. A transformation equation is one that helps us describe an event (such as the position or motion of an object) consistently across multiple frames of reference. Although deriving these transformations can be incredibly complicated we will try to simplify them as far as possible.

Let us say, with reference to the figure above, that some event takes place at . Say a bulb bursts there. The position is at rest with respect to the frame on the right, shown to be moving with some velocity . This is called the primed frame because we use primes and to represent it. There is of course a in the so-called rest frame and a in the primed frame but we will restrict ourselves to two dimensions for now.

Right off the bat we can state that, after some time period , the position of the event is related to from the perspective of an observer in the rest frame as the origin of the primed frame plus the distance the primed frame moved in time . This distance is simply the product of velocity and time or depending on which frame we view it from (also, these have not been expressed vectorially yet). The mathematical description of this is

which is known as a Galilean transformation. The vice versa is also true: .

However, let us say that this is not correct at relativistic speeds. Let us say there is some correction factor that still needs to be applied and call this . Our equations must then become

(1)

and

(2)

That the same gamma factor applies to both is a crucial requirement. This is so because all observers in physics are equivalent; that is to say, the laws of physics must not change for different observers. Our task now is to determine what this gamma is.

We have been careful so far in not making any assumptions about time. Galilean transformations put the time measured by an observer in the unprimed frame as equal to the time measured by an observer in the primed frame. Lorentz transformations, which is what we are formulating now, do not make this assumption. Indeed let us say that the unprimed observer measures at time and the primed observer measures at thereby calling to the fore Einstein’s key idea: that in a vacuum is the speed limit of the universe.

From the simple velocity–time–displacement relationship we have and for our two frames of reference. Let us now multiple the left- and right-hand sides of eq. (1) and (2) together:

It is clear from this that is given by

(3)

This factor is called the Lorentz factor and plays an extremely important role in many relativistic calculations. What it tells us, in effect, is that if two observers in a primed and an unprimed frame, as shown above, measured an event and located it at and respectively in their frames of reference then the coördinates are related as given by eq. (1), (2) and (3).

The length of an object, which we will keep at one dimension for simplicity, is given by two points instead of one: say and marking the two ends of an object of length lying along the *x*-axis in the primed frame.

The length of the object as measured by an observer in the unprimed frame is then related to as follows:

Assuming that the object is small enough that the two time measurements and of the two ends of the object are made simultaneously (at ) we can cancel out the second and fourth terms on the right-hand side of the equation above to get a simpler form:

(4)

where maintaining the same notation but without primes. Note that because the denominator of becomes a quantity less than 1 making the Lorentz factor an increasing quantity, i.e. multiplying a quantity by will either make that quantity larger or (when is incomparable to ) retain its magnitude.

Therefore eq. (4) tells us that the length of an object moving at speeds close to that of light, measured by a stationery observer in the unprimed frame, will appear to have been shortened by a factor of when compared to the length of the object when it was at rest in the unprimed frame. This observation is known as **length contraction**.

The effects of varying length can be seen in terms of time as well but as a reverse phenomenon. On the surface, think of this as another case of the velocity—time—displacement relationship. If the velocity is to be maintained at a constant comparable to then for contracting lengths the times must expand or **dilate**. Of course this is verbal and somewhat a less accurate picture than we can draw using elementary mathematics.

From eq. (1) we can write the time as

which, on substituting from eq. (2) gives us

It is easy to see that

as a result of which we end up with

(5)

Now suppose an arbitrary time interval was measured in the primed frame as (say). We can calculate the same interval as measured in the unprimed frame as which is determined as follows:

which, since the times and are made at the same spatial coördinate, and we have

(6)

is the time as measured in the primed frame. Since gamma, as we stated previously, is an increasing factor, the time measured in the primed frame, i.e. the frame moving at speeds comparable to appears to be longer than in the primed frame.

The takeaway here is that time appears to move at different rates for two observers even if all they are doing is moving relative to each other. All of these are special relativistic effects.

The time dilation in *Interstellar* is not exactly special relativistic, however. Our purpose up to this point was to see that time can run at different rates. Whereas in our previous example it was relative motion, general relativity, along similar lines, tells us that the rate of time can vary even with gravity.

For an object moving close to a massive body,

where is the time measured by the object itself and is that measured by an observer at a great distance away from the massive body. and are the mass and radius of the massive object (the black hole from *Interstellar* for example) while is the universal gravitational constant. Clearly, for extremely massive objects the time interval is sufficiently smaller than , i.e. time slows down in a strong gravitational field.

So it is not that time runs slower in space, rather that it runs slower for the main characters in the film because they are under the influence of a strong gravitational field from a black hole; others further from the black hole experience a much faster rate of time and age longer before the two groups of people meet again. One last point: you may be interested to know that this is really an exaggerated form of a type of relativistic puzzles among which the most famous is the *twin paradox*.

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]]>The post The electric current–water analogy appeared first on Physics Capsule.

]]>Current electricity isn’t really a concept that is tough to understand, and the following analogy should show the reader why. Let us consider a pipe with water. The pipe is the wire. The water, or rather, the particles of water are the charge carriers, the electrons within the material of the wire.

As mentioned, the molecules of water are the charges within the conductor. In that statement, it is tempting to explore the discreteness of electrons and water molecules, but our aim is to simplify the concept. For that reason, we consider current in it’s macroscopic, seemingly continuous form, and the same goes for water.

Both current and water, as per English language, ‘flow’. Both flowing consist of motion of the constituent particles. The flow of current through a wire could easily be thought of as the flow of water through a pipe. In fact, the low resistance that a connecting wire offers to current can be thought of as the friction between the pipe and the water.

Numerically, current is measured as the amount of charge passing through a cross-section of the conductor per unit time. Based on our analogy, we have current as the amount of water, the volume, that passes through a cross section of the pipe in a unit of time. So, our analogy, with respect to current, is complete.

A battery produces a direct current when connected to a complete circuit. Actually, what it does is produce a potential difference, and the charges start flowing within the circuit, from the positive terminal to the negative terminal. It is common knowledge that a flow is always from a region of higher potential(im a more general sense of the word) to lower potential. Heat flows from a region of higher temperature to a region of lower temperature. Temperature is the ‘potential’ for heat.

A body released at a height falls down to the ground. Height from the ground is the ‘potential’ in terms of gravity on Earth. Similarly, Voltage is the potential in case of currents. Since our analogy uses water, our potential has to be height. Think of it this way — a battery is a device that tilts our pipe such that one end is higher than the other, with a machine that pumps water from the lower end back to the upper end. Due to the difference in heights, water flows.

For comparison, the potential difference may be equated to the angle of tilting of the pipe from the horizontal. (Also, for the integrity of the analogy, keep the angle less than . It is the difference in angle that matters here, so at times when numerical comparison is required, one may set a value of how many volts a degree of tilt represents.)

This is fairly simple one. Resistors are a type of component of the electric circuit that resists the flow of current. In terms of potential difference, what a resistor does is decrease the voltage drop across itself. The current does not pass as freely through a resisor as it does through a wire. In our analogy, a resistor is a part of the pipe whose tilt is in the opposite direction to the tilt caused by the battery. As mentioned above, for all comparative purposes, the resistance offered per degree of tilt is our choice. For our analogy to work, it is better to keep between 0 and . The upward(considering that the battery causes a downward tilt,) tilt makes it difficult for the water to flow over that part. Once it passes through that part, the water experiences the same downward tilt as before.

Note that this analogy works as far as we consider direct currents, and it isn’t as complete as to explain more subtle effects, like magnetism, and working of components like an inductance or a capacitor. But, it does help in creating a simple picture of the very basic phenomena. Once the reader acquires enough experience in the topic, this analogy may be thrown away, or developed upon to fit other concepts. This is actually how theories are made: conjecture, experimentation, and modifications or rejctions, based on experiments.

A series connection of components can be pictured exactly how it seems. All components connected one after the other. Consider three resistors connected in series. In our analogy, it is just three upward inclinations, whose tilts depend on the corresponding resistances, one after the other, connected by pipes that have downward inclinations, whose tilt depends on the battery connected.

In this case, the amount of water flowing through all resistors are same, but the voltage drop across each resistor depends on its tilt, or resistance. This is because the current should be the same across every cross-section of the circuit, it has nowhere else to go. On the other hand, the voltage drop is the relative difference of the tilts — downward by the battery and upward by the resistance. It agrees with observation. In a series circuit, current remains the same through out, while the voltage drop varies from component to component.

Now consider three resistors in parallel connection. Imagine that our pipe branches into three tubes, each connected to one resistor, and then joins back as one before reaching the battery terminal. It is important to note that each of the three branches has the same tilt. Quite obvious, because, considered alone, each of them is connected to the same battery. Therefore, as expected, each branch has the same voltage.

Coming to currents, since the amount of water inside the pipe is always a constant, no water can come into the pipe. So, the three branches are fed by fractions of the amount of water in the main pipe. The values of these fractions are decided by the resistors. Greater the resistance, lower is the chance of water going through it. Nature always finds the easier path, though the tougher path is not entirely forgotten.

In parallel connections, the difference in resistances divides the current, while in series, the same difference divides the voltage.

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]]>The post Energy, work and power appeared first on Physics Capsule.

]]>Energy is everywhere, so much so that it has become an overly abused term these days. The same is true, perhaps even more so, of work and power. People refer to everything from the electric current in their homes to their weekend gardening chores as forms of energy, work or power; and they are right to some extent, but not entirely.

The ideas discussed in this article are elementary but extremely important because they will find use elsewhere, in more complex discussions, such as Lagrangian and Hamiltonian mechanics or even in nuclear physics or, most famously, in quantum and condensed matter physics.

Defining energy is harder than it might appear. The joke is that energy, work and power are best defined in terms of each other cyclically, with no start or end. It seems like a safe thing to do (it is) but it is hardly a satisfying approach.

In physics, energy is best defined mathematically rather than verbally. Also, there is no universal equation governing energy but the many forms of energy have their own. Generally speaking, *energy is a property that must be transmitted *to* a system to perform work on it or *by* a system for it to perform some work*.

This is where we encounter some hazy descriptions. What exactly is work? For now, take work to mean exactly what you normally do: gardening, jogging, reading a newspaper, working on your computer, teaching, cycling etc. To do all these, a body (e.g. you or he or she) has to lose *something*, and, if given the same thing, you will be able to do some more of these activities. That *thing* is energy.

In all these examples, specifically, our conversations revolve around a form of energy known as **kinetic energy**. We will better define this in a moment, but first let us see what other type of energies exist: turning on a light bulb is said to involve electric energy; heating a tough of water involves heat or thermal energy; mixing two reactive liquids can involve chemical energy; when the earth pulls you down we say there exists gravitational energy; when a system opposes this gravitational energy it must contain within itself an appropriate amount of energy that it can potentially use elsewhere, which we simply call its potential energy. Likewise there are several different forms of energy that exist mutually or exclusively as the situation demands.

Regardless of what energy we refer to, we always measure energy in joule. However, electron volt and watt too are units that can be used to directly or indirectly imply the involvement of energy, but they can all be written in terms of joule.

*The energy possessed by an object due to its motion is called kinetic energy.* It is defined mathematically, for an object of mass moving with velocity , as

(1)

Clearly, energy is a scalar. (Recall that for a vector its square is a scalar.)

We shall now make a minor detour from energy to discuss what it means to do work. In physics *work is done when a force displaces a mass*. As a result, when either the applied force, the displacement or both are zero, no work is done. Spinning your arms, therefore, means no work is done. Sliding a feather away from you, say, does involve work.

Mathematically,

which means if a force and displacement exist perpendicular to each other (not too common a case) no work is done. More practically, it means that the more the difference is between the directions of the force and displacement, between zero and perpendicularity, the less is the work done.

Think of work and energy as two ways of looking at the same thing. Indeed this is what the **work-energy theorem** tells us: *work is given by the change in energy of a system.* That is,

It is possible to extend this definition to define *positive* and *negative* work, which is generally (in physics) defined as work done *on* and *by* the system respectively. Therefore, positive work sees an increase in energy and negative work sees a decrease in the same.

We can, in fact, use the above two equations to see how eq. 1 comes about:

where is the momentum of the body. Force and momentum are closely related to each other as we have discussed before under Newton’s second law.

Now, observe that

which gives us

just as we said earlier. However, note that the energy expended by a rotating body is different from one exhibiting linear (or approximately linear) motion, but we will put that aside as discussion for another day.

The other important type of mechanical energy we are interested in is potential energy: the energy potentially contained in a body which is at rest or, more scientifically, the energy contained in a body placed inside a field. The field we are all in, the gravitational field of the earth, is what gives us all our most familiar type of energy: gravitational potential energy. We will call it potential energy for short here. This is sometimes also referred to as GPE (arguably wrongly), but that might mean something slightly different, often given in terms of Newton’s universal gravitational constant, so we will not call it GPE here.

The potential energy of a body, like , depends on its mass. However, since the body is explicitly *not* in motion, the does *not* depend on . Instead, it depends on the field that the body is placed in, which, in case of , means the energy must somehow depend on , the acceleration due to gravity.

To see how let us turn to the idea of work again and recall that, in this case, the force acting on the body is simply its weight , which leaves us with

This equation contains a rather interesting term: . If the object is forbidden from moving as far as potential energy is concerned, what is the displacement term doing here? It turns out that this is not so much the displacement caused by the force but, instead, the displacement a force could *potentially* cause if the body moved.

If a force caused the body to move, by how much would it displace the body? This is our term. In case of attractive forces, think of this as the separation between two bodies. A magnet can potentially move an iron nail over a distance between itself and the nail and no more or less. An object kept at a heigh above the earth, say, on a table, can *potentially* be dropped through a displacement of by the earth. Therefore we write the potential energy of a body as

(2)

where is the mass of the body, is the acceleration it experiences due to the earth but just as well due to different factors depending on the system it is in, and is the separation between two bodies in case of attractive force fields. In fact it is in the presence of attractive forces that we will see this equation being used repeatedly although there is no valid reason to treat this as a given.

A term often coupled with work and energy is the term *power*. Like the other two, power is also frequently misused in everyday life. It usually refers to either electric power (although not strictly as a product of current and potential difference) or physical strength.

In physics, *power is the rate of doing work*. Mathematically,

(3)

where is the power associated with the system. The logic is fairly simple: the faster one works, the more energy they expend in a given time, and the more power they have. In mechanics, though, power is of much less importance than energy; in current electricity or electronics, however, power can be an important characteristic describing the capabilities of the device.

Energy, as is evident from eq. 1 the dimension of energy is . Naturally, the same is true of eq. 2 too. The unit of energy is a joule, named after the Englishman, James Prescott Joule, who contributed greatly towards early studies of energy and even defined it. It is also the unit of work.

One joule is defined as *the work done by a force of one newton in displacing a mass through one metre.* It is, therefore, equivalent to one newton-metre.

There are other forms of energy, with other descriptive formulae (e.g. in terms of relativity); there are also alternate definitions of power, as said above, and of the joule (e.g. in terms of electricity as the heat given out when an ampere of current passes through an ohm of resistance for a second). All of these, however, are best discussed in context and it is only basic mechanical energy, work and power that we will concern ourselves with at this point.

Understanding these will help us understand other ideas of classical mechanics. Start with Newton’s laws or Lagrangian mechanics for a better idea of how energy comes into play when we try to understand various systems around us.

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]]>The post The Lagrange equations appeared first on Physics Capsule.

]]>Rewriting the expression for the generalized force,

(1)

We also know from the definition of generalized force,

But from the transformation equations,

(2)

As we mentioned earlier, let us consider conservative forces. These forces are special in the sense that work done by them in moving a mass in space is independent of the path taken and depends only on where you started and where you stopped (much like it is with electric forces). And so, we can express them as a gradient of a potential, or . Also, since, , equation(2) becomes,

And we are left with,

(3)

I.e., if the forces involved are conservative, we can also write the generalized force as the gradient of the potential with respect to the generalized coordinates.

Let us substitute the result (3) into equation (1), we have,

Or,

We might as well combine the last two terms of the equation above (since differentiation is a linear operator) and write,

(4)

If we further choose only potentials that do not depend on velocities, we have . So, no harm is done if we included this in the first term above,

Let’s name the difference , the Lagrangian.

We then can write,

These are the all-famous **Lagrange equations** which form an alternate method of determining the equations of motion (besides the Newtonian method). To put them to use, all you need to know is the total kinetic energy and potential energy of the system. Take the difference of the energies, find the Lagrangian, , and differentiate it according to this equation, and voila, you have your equation of motion! No need to even think of the constraint forces. No need to draw any force diagrams.

But keep in mind that we have derived these equations for forces that are conservative with a potential that is independent of velocities. For example, we can’t use the above form of the equations when forces such as friction or air drag or any other dissipative ones are acting on the body.

Now, focusing on the new function we introduced above – the Lagrangian. Since, it is just the difference of the kinetic and the potential energies, where the kinetic energy is a function of and , and the potential energy of and ; the Lagrangian must be a function of , and .

As we know, there can be ambiguity regarding the potential function chosen, since we are free to chose any point in space as the point of zero potential. So, say, instead of a potential V that you chose, we choose a potential , where is also some function of and . How will the Lagrange’s equation turn out then?

(5)

We can rewrite the second term here as .

And since is a function of and , .

Which implies, .

Hence, equation (5) becomes,

And we get back the Lagrange’s equation,

So, there seems to be no effect on the equations of motion, of adding the function to the Lagrangian. This means that we can construct a number of different Lagrangian functions (differing by this additional function ) all of which give the same, correct equations of motion.

We will next use the Lagrange equations for different classical systems and surprise ourselves with how simply we can obtain the same results with them, as compared with Newton’s way.

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]]>The post The generalized force appeared first on Physics Capsule.

]]>Consider the equation for the D’Alembert principle that we have already derived:

(1)

Now, if the coordinates being used here are responsible for this equation to appear fussy or complicated (in a given problem), we know that we can always switch to other coordinates with the help of the transformation equations between the coordinates. For e.g., if we chose to switch to some generalized coordinates , we have at our rescue the transformation equations,

and so on.

Or, in short hand notation, we shall write them as,

Then the virtual displacement, which just is an infinitesimal variation of the coordinate at a frozen instant of time , can be written as (using the chain rule),

Replacing with this expansion, in equation (1), we can write the virtual work as,

We here define the expression,

(2)

as the components of a “generalized force”.

(Observe that just like generalized coordinates needn’t be measured in meters, the generalized force needn’t be necessarily measured in newtons. But the product of the generalized force and the generalized coordinates must be in joules or any equivalent unit of work.)

With this, we can comfortably define virtual work as a product of the generalized forces and the generalized coordinates.

(3)

We’ve successfully expressed the first term of equation (1) in terms of the generalized coordinates. We have to do the same with the second term .

Firstly, since by definition of momentum , we have, (assuming the masses of the particles don’t change with time). Then, the second term of equation (1) shall be rewritten as,

But we already know what to replace with. So,

(4)

Now, say we want to calculate the time derivative:

Oh, the first term on the right side here seems to have something to do with the right side of our equation (4).

Therefore, we can rewrite equation (4) as,

(5)

The second term in the parenthesis on the right side here contains a derivative of first with respect to and then with respect to . We could as well interchange the order of these differentiations and write,

Where we have used the definition of the Cartesian velocity components: ; since the ‘s are a function of the ‘s. This equation here also implies that . Now, putting all these results into equation (5), we have,

(6)

We hope you still are aware of where we are headed to. We have been rewriting equation (1). So, putting in the results of equation (3) and equation (6) into equation (1), we have,

(7)

Closely observe the first term in the parenthesis here, does it not look like the expansion,

Similarly,

So, we shall rewrite equation (7) as,

Here, is just the famous classical kinetic energy , of the masses. Hence, we may write,

Another way of writing this is,

The virtual displacement was chosen arbitrarily (nothing particular about it was assumed), and so for the above equation to hold for a non-zero displacement, the term in the parenthesis must vanish:

Or,

(8)

We have here a recipe for obtaining the equation of motion of any object moving due to a force. Knowing the forces and using the relation (2), , we can know the generalized forces. Then all that remains is for us to calculate the kinetic energy of the object and put it all in the above equation (8), and we know how the object moves.

In the following article, we will see how the concept of the generalized force we understood here, can be used to deduce a set of marvelous equations called the Lagrange equations, which form an alternate explanation of object motions.

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]]>The post Virtual displacement and the D’Alembert principle appeared first on Physics Capsule.

]]>We call such a change in position in zero time (by an infinitesimal, teeny-weeny length), the **virtual displacement** of the object. But we must be careful: we here allow ourselves to imagine motion in zero time, but retain the limitations of the motion that exist in real time. That is, if an object’s motion is confined in some manner, for e.g., a book placed on a table top is free to move anywhere on or above the table, but not “into” the table; this restriction will be carried over to virtual displacement as well. And in a more general set up where the forces are constantly changing with time, we consider the forces only at the instant at which the virtual displacement happens.

Now to cause a change in the position of any object, in zero time or real time, requires a force (assuming the object was at rest initially). In fact, the very concept of force was in the first place introduced to account for a cause for displacements. Hence here, a force causing a virtual displacement will lead to a work done, a **virtual work done**. If is the virtual displacement caused by a force , on the particle of a system, the virtual work will be

But if the net force on each particle is zero, then each particle is in mechanical equilibrium, and the virtual work done on each particle will be zero. The same argument holds for the entire system:

Now, the force can be from two sources – it could be an externally applied force or it could be a force of constraint (the force that does not allow the book to move “into” the table). Therefore, we have,

And the total virtual work done will be (for this equilibrium case),

Or,

We very well know that work done is zero when the applied force is perpendicular to the direction of displacement. So, we could get rid of the second term above, by demanding that the forces of constraint be always perpendicular to the direction of the virtual displacement (i.e., we are dealing only with cases that satisfy this condition). So, forget frictional forces. Also, we could be dealing with constraints that are changing with time. Imagine an ant on a balloon into which air is being blown constantly, causing it to swell. The constraint for the motion of the ant here, is changing with time. But, when we started speaking of virtual displacements, we froze time, didn’t we? And so, we are speaking of the motion of the ant when the balloon is at a specific stage during its expansion. And in such a scenario, the force of constraint (the force that doesn’t allow the ant to simply diffuse into the balloon) will always be perpendicular to the (virtual) displacement of the ant.

With the contribution from the constraint forces gone, the total virtual work will be,

Looking at this equation, it might seem like, . But no. From the physical perspective, is the external force experienced by the particle. When we began this discussion, we only demanded that the total force on each particle be zero, not that the external force (which is only a part of the total force) be zero. Even though we have arranged for the contribution of the constraint forces to the virtual work, to vanish, we still do have the constraint forces. And these in fact connect the coordinates .

For e.g., if say our system had just two particles, the virtual work of the system would be (under equilibrium),

But the coordinates and could be connected via the constraint equations. Say, we had the relation as the constraint, then , and we would have

Or,

Since was chosen arbitrarily, this would lead to the relation,

As long as this relation holds, it is not necessary that and be zero. More generally, as we stated before, , in general, because the are not independent and are in fact connected via the constraint equations.

Newton said, force is the rate of change of momentum. If you observe an object’s momentum change, there’s always a force behind it. (Note that we are now releasing ourselves from the clutches of equilibrium.) One of the most celebrated equations in Physics,

We could rephrase this statement as: if is the force that causes a change in momentum of the object, then we would require a force of magnitude in the opposite direction, to keep the object in equilibrium. So, instead of imagining an object changing its momentum under the influence of a force, we could as well imagine the object at equilibrium, under the influence of two equal and opposite forces and . So, our net force here is .

We can therefore write, for the virtual work done, under this balanced force,

Or, speaking of a system of particles, the net virtual work done on the system will be,

Again, the old argument: the force here is .

And so,

Demanding again that the constraint forces be perpendicular to the virtual displacements, we are left with,

This is the famous D’Alembert’s principle. It gives us the condition under which a given system will remain at equilibrium.

In our next article, we will extend this principle into generalized coordinates and introduce concepts such the generalized force, which will ultimately help us understand motions of objects.

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]]>The post The postulate of equal a priori probabilities appeared first on Physics Capsule.

]]>Say, our macroscopic system is something like a gas inside a sealed box. We wish to describe this gas quantitatively. The most straightforward way to do this is to count the number of particles forming the gas and specify the positions and momenta of each particle, at a specific instant of time. This set of values of the positions and momenta of the gas particles, is what we call the **state of the system**. If we observe the system at a later instant, the positions and momenta of the particles would have changed, in general, and hence we’d say that the state of the system has changed.

With the system set up, let us now make our aim more specific: We want to find the probability of finding our system in a specific state, out of the literally infinite possibilities. Before you scoff us off saying that such a probability would essentially be equal to zero – it’d be like finding the probability of catching a specific mosquito out of the billions that exist out there. But you must agree to the fact that the problem becomes much more solvable if we were to find the probability of finding a mosquito belonging to a specific species, rather than hunt a particular individual. Therefore, we impose certain constraints on our system. Our system has a volume , containing number of particles and the energy of the system lies anywhere in the margin . With these impositions, the number of possible states the system can exist in, reduces.

Now, if you look at the problem from a slightly different perspective, you will appreciate that calculating the probability of our system having a specific state (under the given constraints) is equivalent to calculating the probability of picking a single system, out of a large number of systems, in which each system is in a unique state. I.e., instead of imagining a large number of possibilities for a single system, imagine a large number of systems, each having one possibility. With the mosquito analogy, here you are, instead of finding the number of diseases a single mosquito can spread, you are considering a large number of mosquitoes and assuming that each mosquito spreads only one kind of disease. Then the probability of a particular disease being spread from a single bite of your mosquito, will be equivalent to the probability of any one of the mosquitoes of the large group of mosquitoes, biting (assuming that a single bite will certainly inflict a particular disease!).

Now, this large number of systems isn’t something real. They are mere mental copies of the system that we’re trying to study. All these systems are identical in composition but each has a different state (under the constraints). We call such an imaginary cluster of systems, an **ensemble**. Our system, that we set out to study, is one of these.

We’ve set up this extraordinary setting of systems, only to calculate the probability of finding our system in one of the states that obey the constraints. As explained above, this will be equal to calculating the probability of picking a system from the ensemble, that has any of the required states (under constraints). So, we simply calculate the total number of systems in the ensemble that have these required states and divide by the total number of systems present in the ensemble.

One important detail is that as we calculate the probability, our original system is in thermal equilibrium. Meaning, its “macrostate” does not depend on the passage of time (macrostates involve macro properties such as temperature and pressure) .

Finally, the big question – will the probability of picking a system from the ensemble be the same as that of picking any other? In other words, is it equally likely that our system be in a specific state or in any other, obeying the assumed constraints? The answer is a simple yes. How? Try to answer, why is it that when a coin is tossed randomly, it is equally likely that you get a head or a tail? You’d say that it is common knowledge that a head or a tail are equally likely to turn up. We say the same here: it is obvious or using the specific phrase “a priori”, to conclude that the probability of finding a system in a state is equal to the probability of finding it in any other state of the ensemble. There’s no bias; nothing that would lead us to suppose that one of the states is more likely to be found than another. All states have an equal chance of turning up. Hence, the equal a priori probabilities.

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]]>The post The origins of life appeared first on Physics Capsule.

]]>Rewind 4.5 billion years: the solar system has just formed from a cloud of dust that collapsed. Most of the collapsed mass is at the centre of the system and has formed the Sun, around which revolve hundreds of rocky masses. These are constantly colliding, sometimes clinging to one another and forming larger rocks, and at other times shattering into tinier ones. Such repeated processes eventually lead to the formation of larger masses called planets. Our Earth was one of them, but it was nothing like how it is today.

The surface of the earth was but a sea of molten rock. It’d take some time until the surface cools and solidifies. Even after the cooling, there’s a lot of heat trapped beneath it, which often erupts out in the form of volcanoes. The gases from these eruptions will form the first atmosphere of earth. But still, there’s no water, no oxygen, and no chance for life to thrive.

It was the icy meteors which entered the earth from outer space that introduced water into our atmosphere. With clouds formed, there was rain. And the earth harbored water on itself for the first time. But the presence of water only facilitates life; it does not necessarily imply the origin of life. Then how exactly did life begin?

As we said before, we don’t know how it all began. Different people have guessed different explanations. Some say it was the lightning bolts from the skies, some say it was the deep sea vents underneath. Some even say that the comets that brought the water also brought the first microbes and life evolved. And some say it all happened in the most natural manner – inch by inch, molecules got more and more complex, with repeated reactions, going from single atoms to single cells to complex multicellular organisms capable of reproduction.

The bacteria that so formed turned the sunlight and the carbon dioxide that was available to them into oxygen (a process we now call photosynthesis). They produced oxygen in such large amounts, that newer life forms that evolved thereafter, based their existence on the consumption of oxygen. With the stage now set, life evolved, becoming increasingly advanced, and from what began as the complexity of our universe came about a system just as complex within every single one of us.

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]]>The post Laws of reflection of light appeared first on Physics Capsule.

]]>When you look at yourself in the mirror every morning, you naturally begin examining your appearance without giving a second thought to the question of how exactly you are able to see yourself in the mirror in the first place. A more careful inspection of what’s happening when you’re looking at your face in the mirror, can be quite fascinating. Firstly, to see yourself in the mirror, the room/place you’re in must be fairly lit. This external light first falls on your face and is reflected in different directions. Let’s consider the light that is reflected in the direction of the mirror – this light again undergoes reflection at the mirror and the reflected light is what enters your eyes and gives you a perception of your appearance. The reflection of light at the mirror happens according to certain rules. There are basically two laws.

Speaking only in terms of single rays: the ray that is incident on the mirror, the ray that is reflected back, and the “normal” to the mirror surface, all lie in the same plane.

Recall that a plane can be uniquely defined by two lines – give us two lines and we will tell you the plane in which they lie. And with the plane defined this way, we can very easily decide whether or not a given third line lies in the same plane. In stating the first law, we have three “lines” – the incident ray, the reflected ray and the normal. While the incident and reflected rays are very much real, the normal is a geometric construction, an imaginary line drawn for our reference. It is drawn perpendicular to the surface of the mirror, passing through the point at which the incident light ray is incident on the mirror (see the image below).

Now, what the rule says is, when a light ray is incident on the plane mirror, it forms a unique plane with the normal. With this plane defined, now the reflected ray has to lie in the same plane (the direction in which the reflected ray can travel is confined to this plane).

The angle of incidence equals the angle of reflection ()

What this law says is that the reflected ray is not just confined to the plane of the incident ray and the normal, but also is confined to travel in a particular direction after reflection. What this direction is, is ascertained by measuring the angles at which the incidence and the reflection happen. The angle of incidence is the angle formed between the incident ray and the normal. Similarly, the angle of reflection is the angle formed between the reflected ray and the normal. Therefore, the law 2 says that the reflected ray gets reflected at a particular angle with the normal which is equal to the angle made by the incident ray with the normal.

In conclusion, how a ray gets reflected from a plane mirror completely depends on how the incident ray is incident. Both, the plane in which the reflection must happen and the angle at which the reflected ray must emerge, are decided by the direction from which the incident ray is incident on the mirror.

An important special case is when the angle of incidence equals zero degrees. This is the case of “normal incidence” of light. As the second law says, the angle of reflection will also be zero. Therefore, the light ray falling normally at a plane mirror, reflects back along the same path that it came through.

And as you go on increasing the angle of incidence from zero, the angle of reflection too increases, and at every instance will be equal to the incidence angle.

Up next, we’ll see how reflection of light happens at mirrors that may not be plane, and may be curved.

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]]>The post Conservation laws: mass, energy and momentum appeared first on Physics Capsule.

]]>All principles of conservation say more or less the same thing, that the observable being conserved neither goes anywhere nor appears out of the blue. The law of conservation of mass says the same:

Quite simple. It has a depth of meaning within it. All mass in this universe is conserved. No new mass can be created (meaning, no magic), and, conversely, the mass already present can’t be destroyed. The meaning of ‘destruction’ here is to be understood properly. Sure, you can break a chalk piece, but (classically) the number of chalk molecules before and after the breakage, on your hands, on the two pieces, and those that fell on the floor, remain the same.

The latter part of the law allows for the phase transitions of matter and for complex phenomena like radioactive decay. Also, from jewelery to furniture, anything that involves making items by melting and molding and such, is an example, and a proof of this law. To make an iron bar, one has to mine the ferrous ore, get iron from it, melt it into a liquid, pour it into a mold, and let it cool. No iron ore, no iron bar. That’s the iron rule, if you will excuse the pun.

A subtler, more restrictive condition also exists. Mass is localized, that is, it occupies a limited amount of space. It has to have a head and tail, so to speak, at some point. Mass retains it existence at all points of time. If a ball has to go from point A to point B, it has to go through some path between A and B. It has to *travel* the distance between. The ball will not suddenly disappear from the South Pole and reappear – with or without a time lag – on the North Pole.

The same Law that applies for mass applies for energy as well.

Simple, again. The total amount of energy in this universe is conserved. What the universe had at the start of time, it has now, and it will have in the future. The second sentence is what clears the huge numbers of doubts that come to mind because of the first. Most natural displays of motion are instances of conversion of potential energy into kinetic energy. A spring jumping when the force pressing it down is released, anything falling from a higher place to a lower place, an arrow flying from a bow, etc.

*Potential energy* is an apt name. A body that possesses this energy has the *potential* to do something – jump or fall down or whatever the case may be. Let us go about it one by one:

- When a spring is pressed, all the energy applied to it gets stored as potential energy, simply, , where is the co-efficient of stiffness of the spring and is the length by which it is compressed. When the pressure is released, there is nothing holding back the spring, and all the energy stored is converted into kinetic energy, and the spring jumps.
- When something is lifted up, work is done against the force of gravity. All this work done is kept stored in the body as potential energy. When the object then has a chance of falling down, the potential energy slowly transforms itself into kinetic energy. This transformation goes on till either the object’s fall is broken or it reaches terminal velocity, by which time its potential energy is converted into kinetic energy, lost as friction etc. This increase in kinetic energy is basically increase in , and in usual cases, since the mass m is a constant, it is the velocity, or rather, that increases. This is another way of looking at why something accelerates while falling down, as opposed to the use of g.
- As the arrow is pulled back, the bowstring is stretched taut, storing potential energy into it. When released, all of this potential energy is converted into kinetic energy as the string tries to get back to its original state. This violent jerk is passed on to the arrow, shooting it forward and off the bow.

In all these cases, one point is of importance. The kinetic energy that the body acquires is derived from the potential energy that it had, and nothing else. This implies that, in any case, the kinetic energy is equal to, or less than the potential energy, never more than it. In mathematical terms, the total energy of the body, , is equal to the sum of its kinetic and potential energies,

This implies the law of conservation – without any external source of energy, the LHS of the equation remains a constant, and, in turn, any change in the kinetic energy forces an equal negative change in the potential energy.

Apart from this, there are other occurrences that are not simple changes from stored energy to moving energy. Riding a cycle, for instance. The rider uses his weight and muscular energy to rotate the pedal, which, using a system of gears, gives rotational energy to the wheel. The wheel rotates, and by friction between the tyres and the road, the cycle goes forward. In other words, a bicycle is a machine that allows you to convert muscular energy to rotational energy and that to kinetic energy. It wouldn’t work if the rider does not have enough strength in him.

One of the most famous equations in physics, though many do not grasp its true meaning, is Einstein’s mass-energy equivalence, , where c is the speed of light. This is not to be taken at face value. Indeed, a coin or a button has nothing to do with the speed of light. What the equation means to say is that, *if* a coin of mass 2 grams was completely converted into energy, the result would be joule. Try to write down the number with all its zeroes and you realise how big it is.

Why is the equation so important ? That equation is what explains the energy released during a nuclear reaction, for example. During a nuclear reaction, it was observed that there is a slight difference between the sum of masses of the reactants and that of the products. This slight defect, , is converted to energy, and is numerically given by this equation, so, it is useful in reactors.

Consider another case when matter becomes energy: when matter and antimatter meet. Every particle has an antiparticle, like a positron for an electron and an antiproton for a proton, that is same in all respects except for the fact that it carries the opposite charge. Matter that is made of these antiparticles is aptly named antimatter. When a particle and its antiparticle meet, they annihilate each other and what is left is a lot of energy, numerically given by , where M is a particle’s mass.

But all of the above seemingly go against the law of conservation of mass. Solid mass seems to be destroyed! That’s where the equivalence of mass and energy plays its role. Since both independently have their own conservation laws, it is quite called for to have a law of conservation that takes both into account. Mass can become energy, and in some cases, vice versa. So it is not exactly destroyed, and we have

Now we have a general idea of what a conservation law says: ‘No magic allowed’. Everything that happens can be accounted for. Up next, we talk of momentum.

Momentum is defined from Newton’s laws of motion, so it seems apt to refer back to them when we try to understand where the conservation laws come from. For example, Newton’s First law says,

And the Second law gives a hint of what the external unbalanced force does,

Or, mathematically,

where is the momentum of the body. The First Law by itself is the Law of conservation of momentum. The Second is invoked just for a mathematical proof. In the case that the external force is non-existent, we have . That is, in the absence of external force, the momentum of the body does not change with time. is a constant with respect to time.

There are subsections of momentum conservation. Translational motion has rotational analogues. Angle for displacement, angular frequency for velocity, and so on. The analogue for force is torque, , and that for linear momentum, , is angular momentum, . Both linear and angular momenta are separately conserved in the absence of linear force and torque respectively. Momentum is conserved, no magic allowed.

Now to clear a doubt that might occur to you. When studying collisions, we are taught that in inelastic collisions, the momentum is not conserved, since the colliding bodies stick together. Whether this is right or not is a matter of perspective. But it does not violate the law of conservation. The reason for this is that in a collision, there is a non-zero external force involved. Our law is defined when there is no such thing. The two situations are mutually exclusive. So the next time you find yourself in a situation and seem to be missing mass, energy or momentum, know that they are always conserved. They are still out there and you will find them if you look closely.

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