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Mass in special relativity

Massinspecial relativity incorporates the general understandings from the concept of mass-energy equivalence. Added to this concept is an additional complication resulting from the fact that "mass" is defined in two different ways in special relativity: one way defines mass ("rest mass" or "invariant mass") as an invariant quantity which is the same for all observers in all reference frames; in the other definition, the measure of mass ("relativistic mass") is dependent on the velocity of the observer.

The term mass in special relativity usually refers to the rest mass of the object, which is the Newtonian mass as measured by an observer moving along with the object. The invariant massis another name for the rest mass of single particles. The more general invariant mass (calculated with a more complicated formula) loosely corresponds to the "rest mass" of a "system." Thus, invariant mass is a natural unit of mass used for systems which are being viewed from theircenter of momentum frame, as when any closed system (for example a bottle of hot gas) is weighed, which requires that the measurement be taken in the center of momentum frame where the system has no net momentum. Under such circumstances the invariant mass is equal to the relativistic mass (discussed below), which is the total energy of the system divided by c (the speed of light) squared.

The concept of invariant mass does not require bound systems of particles, however. As such, it may also be applied to systems of unbound particles in high-speed relative motion. Because of this, it is often employed in particle physics for systems which consist of widely separated high-energy particles. If such systems were derived from a single particle, then the calculation of the invariant mass of such systems, which is a never-changing quantity, will provide the rest mass of the parent particle (because it is conserved over time).

Despite the convenience that the invariant mass is the same as the total energy of the system (divided by c2) in the center of momentum frame, the invariant mass of systems (like the rest mass of single particles) is also the same quantity in all inertial frames. Thus, it cannot be destroyed, and is conserved, so long as the system is closed. (In this case, "closure" implies that an idealized boundary is drawn around the system, and no mass/energy is allowed across it).

The term relativistic mass is also sometimes used. This is the sum total quantity of energy in a body or system (divided by c2). As seen from the center of momentum frame, the relativistic mass is also the invariant mass, as discussed above (just as the relativistic energy of a single particle is the same as its rest energy, when seen from its rest frame). For other frames, the relativistic mass (of a body or system of bodies) includes a contribution from the "net" kinetic energy of the body (the kinetic energy of the center of mass of the body), and is larger the faster the body moves. Thus, unlike the invariant mass, the relativistic mass depends on the observer's frame of reference. However, for given single frames of reference and for closed systems, the relativistic mass is also a conserved quantity.

Although some authors present relativistic mass as a fundamental concept of the theory, it has been argued that this is wrong as the fundamentals of the theory relate to space-time. There is disagreement over whether the concept is pedagogically useful. The notion of mass as a property of an object from Newtonian mechanics does not bear a precise relationship to the concept in relativity.

For a discussion of mass in general relativity, see mass in general relativity. For a general discussion including mass in Newtonian mechanics, see the article on mass.

Terminology

If a stationary box contains many particles, it weighs more in its rest frame, the faster the particles are moving. Any energy in the box (including the kinetic energy of the particles) adds to the mass, so that the relative motion of the particles contributes to the mass of the box. But if the box itself is moving (its center of mass is moving), there remains the question of whether the kinetic energy of the overall motion should be included in the mass of the system. The invariant mass is calculated excluding the kinetic energy of the system as a whole (calculated using the single velocity of the box, which is to say the velocity of the box's center of mass), while the relativistic mass is calculated including invariant mass PLUS the kinetic energy of the system which is calculated from the velocity of the center of mass.

Relativistic mass and rest mass are both traditional concepts in physics, but the relativistic mass corresponds to the total energy. The relativistic mass is the mass of the system as it would be measured on a scale, but in some cases (such as the box above) this fact remains true only because the system on average must be at rest to be weighed (it must have zero net momentum, which is to say, the measurement is in its center of momentum frame). For example, if an electron in a cyclotron is moving in circles with a relativistic velocity, the weight of the cyclotron+electron system is increased by the relativistic mass of the electron, not by the electron's rest mass. But the same is also true of any closed system, such as an electron-and-box, if the electron bounces at high speed inside the box. It is only the lack of total momentum in the system (the system momenta sum to zero) which allows the kinetic energy of the electron to be "weighed." If the electron is stopped and weighed, or the scale were somehow sent after it, it would not be moving with respect to the scale, and again the relativistic and rest masses would be the same for the single electron (and would be smaller). In general, relativistic and rest masses are equal only in systems which have no net momentum and the system center of mass is at rest; otherwise they may be different.

The invariant mass is proportional to the value of the total energy in one reference frame, the frame where the object as a whole is at rest (as defined below in terms of center of mass). This is why the invariant mass is the same as the rest mass for single particles. However, the invariant mass also represents the measured mass when the center of mass is at rest for systems of many particles. This special frame where this occurs is also called the center of momentum frame, and is defined as the

Velocity

In physics, velocity is the measurement of the rate and direction of change in position of an object. It is a vectorphysical quantity; both magnitude and direction are required to define it. The scalarabsolute value (magnitude) of velocity is speed, a quantity that is measured in meters per second (m/s or ms−1) when using the SI (metric) system.

For example, "5 meters per second" is a scalar and not a vector, whereas "5 meters per second east" is a vector. The average velocity v of an object moving through a displacement ( \Delta \mathbf{x}) during a time interval ( \Delta t) is described by the formula:

\mathbf{\bar{v}} = \frac{\Delta \mathbf{x}}{\Delta t}.

The rate of change of velocity is acceleration– how an object's speed or direction changes over time, and how it is changing at a particular point in time.

Equation of motion

The velocity vector v of an object that has positions x(t) at time t and x(t + \Delta t) at time t + \Delta t, can be computed as the derivative of position:

\mathbf{v} = \lim_{\Delta t \to 0} \over \Delta t}={\mathrm{d}\mathbf{x} \over \mathrm{d}t}.

Average velocity magnitude is always smaller than or equal to average speed of a given particle. Instantaneous velocity is always tangential to trajectory. Slope of tangent of position or displacement time graph is instantaneous velocity and its slope of chord is average velocity.

The equation for an object's velocity can be obtained mathematically by evaluating the integral of the equation for its acceleration beginning from some initial period time t_0 to some point in time later t_n.

The final velocity v of an object which starts with velocity u and then accelerates at constant acceleration a for a period of time \Delta t is:

\mathbf{v} = \mathbf{u} + \mathbf{a} \Delta t.

The average velocity of an object undergoing constant acceleration is \tfrac {(\mathbf{u} + \mathbf{v})}{2}, where u is the initial velocity and v is the final velocity. To find the position, x, of such an accelerating object during a time interval, \Delta t, then:

\Delta \mathbf{x} = \frac {( \mathbf{u} + \mathbf{v} )}{2}\Delta t.

When only the object's initial velocity is known, the expression,

\Delta \mathbf{x} = \mathbf{u} \Delta t + \frac{1}{2}\mathbf{a} \Delta t^2,

can be used.

This can be expanded to give the position at any time t in the following way:

\mathbf{x}(t) = \mathbf{x}(0) + \Delta \mathbf{x} = \mathbf{x}(0) + \mathbf{u} \Delta t + \frac{1}{2}\mathbf{a} \Delta t^2,

These basic equations for final velocity and position can be combined to form an equation that is independent of time, also known as Torricelli's equation:

v^2 = u^2 + 2a\Delta x.\,

The above equations are valid for both Newtonian mechanics and special relativity. Where Newtonian mechanics and special relativity differ is in how different observers would describe the same situation. In particular, in Newtonian mechanics, all observers agree on the value of t and the transformation rules for position create a situation in which all non-accelerating observers would describe the acceleration of an object with the same values. Neither is true for special relativity. In other words only relative velocity can be calculated.

In Newtonian mechanics, the kinetic energy (energy of motion), E_K, of a moving object is linear with both its mass and the square of its velocity:

E_{K} = \begin{matrix} \frac{1}{2} \end{matrix} mv^2.

The kinetic energy is a scalar quantity.

Escape velocityis the minimum velocity a body must have in order to escape from the gravitational field of the earth. To escape from the Earth's gravitational field an object must have greater kinetic energy than its gravitational potential energy. The value of the escape velocity from the Earth's surface is approximately 11100 m/s.

Relative velocity

Relative velocity is a measurement of velocity between two objects as determined in a single coordinate system. Relative velocity is fundamental in both classical and modern physics, since many systems in physics deal with the relative motion of two or more particles. In Newtonian mechanics, the relative velocity is independent of the chosen inertial reference frame. This is not the case anymore with special relativity in which velocities depend on the choice of reference frame.

If an object A is moving with velocity vectorv and an object B with velocity vector w, then the velocity of object A relative to object B is defined as the difference of the two velocity vectors:

\mathbf{v}_{A\text{ relative to }B} = \mathbf{v} - \mathbf{w}

Similarly the relative velocity of object B moving with velocity w, relative to object A moving with velocity v is:

\mathbf{v}_{B\text{ relative to }A} = \mathbf{w} - \mathbf{v}

Usually the inertial frame is chosen in which the latter of the two mentioned objects is in rest.

Scalar velocities

In the one dimensional case, the velocities are scalars and the equation is either:

\, v_{rel} = v - (-w), if the two objects are moving in opposite directions, or:
\, v_{rel} = v -(+w), if the two objects are moving in the same direction.

Polar coordinates

In polar coordinates, a two-dimensional velocity is described by a radial velocity, defined as the component of velocity away from or toward the origin (also known as velocity made good), and an

Harmonic mean

In mathematics, the harmonic mean (sometimes called the subcontrary mean) is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired.

The harmonic mean H of the positive real numbers x1, x2, ..., xn > 0 is defined to be

H = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}} = \frac{n}{\sum_{i=1}^n \frac{1}{x_i}} = \frac{n \cdot \prod_{i=1}^n x_i }{ \sum_{j=1}^n \frac{\prod_{i=1}^n x_i}{x_j}}.

Equivalently, the harmonic mean is the reciprocal of the arithmetic mean of the reciprocals. From the third formula in the above equation it is more apparent that the harmonic mean is related to the arithmetic and geometric means.

Relationship with other means

The harmonic mean is one of the three Pythagorean means. For all positive data sets containing at least one pair of nonequal values, the harmonic mean is always the least of the three means, while the arithmetic mean is always the greatest of the three and the geometric mean is always in between. (If all values in a nonempty dataset are equal, the three means are always equal to one another; e.g. the harmonic, geometric, and arithmetic means of {2, 2, 2} are all 2.)

It is the special case M−1 of the power mean.

Since the harmonic mean of a list of numbers tends strongly toward the least elements of the list, it tends (compared to the arithmetic mean) to mitigate the impact of large outliers and aggravate the impact of small ones.

The arithmetic mean is often mistakenly used in places calling for the harmonic mean. In the speed example below for instance the arithmetic mean 50 is incorrect, and too big.

The harmonic mean is related to the other Pythagorean means, as seen in the third formula in the above equation. This is noticed if we interpret the denominator to be the arithmetic mean of the product of numbers n times but each time we omit the jth term. That is, for the first term we multiply all n numbers but omit the first, for the second we multiply all n numbers but omit the second and so on. The numerator, excluding the n, which goes with the arithmetic mean, is the geometric mean to the power n. Thus the nth harmonic mean is related to the nth geometric and arithmetic means.

Weighted harmonic mean

If a set of weights w_1, ..., w_n is associated to the dataset x_1, ..., x_n, the weighted harmonic mean is defined by

\frac{\sum_{i=1}^n w_i }{ \sum_{i=1}^n \frac{w_i}{x_i}}.

The harmonic mean as defined is the special case where all of the weights are equal to 1, and is equivalent to any weighted harmonic mean where all weights are equal.

Examples

In physics

In certain situations, especially many situations involving rates and ratios, the harmonic mean provides the truest average. For instance, if a vehicle travels a certain distance at a speed x (e.g. 60 kilometres per hour) and then the same distance again at a speed y (e.g. 40 kilometres per hour), then its average speed is the harmonic mean of x and y (48 kilometres per hour), and its total travel time is the same as if it had traveled the whole distance at that average speed. However, if the vehicle travels for a certain amount of time at a speed x and then the same amount of time at a speed y, then its average speed is the arithmetic mean of x and y, which in the above example is 50 kilometres per hour. The same principle applies to more than two segments: given a series of sub-trips at different speeds, if each sub-trip covers the same distance, then the average speed is the harmonic mean of all the sub-trip speeds, and if each sub-trip takes the same amount of time, then the average speed is the arithmetic mean of all the sub-trip speeds. (If neither is the case, then a weighted harmonic mean or weighted arithmetic mean is needed.)

Similarly, if one connects two electrical resistors in parallel, one having resistance x (e.g. 60Ω) and one having resistance y (e.g. 40Ω), then the effect is the same as if one had used two resistors with the same resistance, both equal to the harmonic mean of x and y (48Ω): the equivalent resistance in either case is 24Ω (one-half of the harmonic mean). However, if one connects the resistors in series, then the average resistance is the arithmetic mean of x and y (with total resistance equal to the sum of x and y). And, as with previous example, the same principle applies when more than two resistors are connected, provided that all are in parallel or all are in series.

In other sciences

In Information retrieval and some other fields, the harmonic mean of the precision and the recall is often used as an aggregated performance score: the F-score (or F-measure).

An interesting consequence arises from basic algebra in problems of working together. As an example, if a gas-powered pump can drain a pool in 4 hours and a battery-powered pump can drain the same pool in 6 hours, then it will take both pumps (6 Â· 4)/(6 + 4), which is equal to 2.4 hours, to drain the pool together. Interestingly, this is one-half of the harmonic mean of 6 and 4.

In hydrology the harmonic mean is used to average hydraulic conductivity values for flow that is perpendicular to layers (e.g. geologic or soil). On the other hand, for flow parallel to layers the arithmetic mean is used.

In sabermetrics, the Power-speed number of a player is the harmonic mean


From Yahoo Answers

Question:What benefits are there for using standard devitaion, over the absolute deviation? The absolute deviation seems much more intuitive (taking the average of the deviation), but since standard deviation is much more common, I assume there are benefits of taking a root mean square deviation.

Answers:Here is a little introduction from a paper. Bottom line, you have a good question, and the common use of variance is due to historical reasons. But there are also theoretical reasons to do so. Revisiting a 90-year-old debate: the advantages of the mean deviation Stephen Gorard Department of Educational Studies, University of York, email: sg25@york.ac.uk Paper presented at the British Educational Research Association Annual Conference, University of Manchester, 16-18 September 2004 Introduction This paper discusses the reliance of numerical analysis on the concept of the standard deviation, and its close relative the variance. Such a consideration suggests several potentially important points. First, it acts as a reminder that even such a basic concept as standard deviation , with an apparently impeccable mathematical pedigree, is socially constructed and a product of history (Porter 1986). Second, therefore, there can be equally plausible alternatives of which this paper outlines one the mean absolute deviation. Third, we may be able to create from this a simpler introductory kind of statistics that is perfectly useable for many research purposes, and that will be far less intimidating for new researchers to learn (Gorard 2003a). We could reassure these new researchers that, although traditional statistical theory is often useful, the mere act of using numbers in research analyses does not mean that they have to accept or even know about that particular theory. for your question, see the paragraph " Why do we use the Standard Deviation?" http://www.leeds.ac.uk/educol/documents/00003759.htm

Question:I really don't understand how to calculate this. I'll give you an example, so that maybe you can help me with it. I just need a formula or a sample of how to do it or something. Four consecutive measurements of a sample of lithium sulfide give the values of 3.45g, 3.44g, 3.44g, and 3.43g. What are the absolute deviations of each measurement and how should you express the final answer to indicate the precision of you answer? My teacher said the formula for it was... Absoute deviation=absolute difference between each measurement and the average I don't really know what he meant by this. (absolute difference between each measurement?)

Answers:he meant absolute deviation is just | measurement1 - average |, | measurement2 - average |, etc... the | | means the "absolute value", which means make your final answer as a positive number only. Your average = 3.44 so your absolute deviations for each measurements are: | 3.45 - 3.44 | = 0.01 | 3.44 - 3.44 | = 0.00 | 3.43 - 3.44 | = 0.01 your answer is always expressed by the average +/- precision or error. So your final answer should be 3.44 +/- 0.01 because 0.01 is your "biggest possible error" or "least precise"

Question:In an experiment, we weighed 10 coin samples and separated the data into two data sets. We have computed for the ff: *mean *standard Deviation *Relative Standard Deviation *Range *Relative range *Confidence Limit (95% confidence level) Now, THE QUESTIONS............ 1. What is the significance of computing for the mean, Standard Deviation, Confidence Limits and the Q-Test? 2. Which is the most useful parameter for illustrating measures of precision?

Answers:/ mean shows the average value of the sample and s.d. shows the spread of the data around the mean,i.e. whether the data is extreme or not.

Question:In a random sample of 700 refreshment-dispensing machines, it is found that an average of 8.1 ounces is dispensed with a standard deviation of 0.75 ounce. 1. Find the standard error of the mean. Round to nearest 10 thousandths. 2. Find the probability that the mean of the population will be less than 0.085 ounce from the mean of the sample. Round to tenths place 3. Find the probability that the true mean is between 8.157 and 8.185. Round to nearest tenths place Please explain Thank you!! :) Will do best answer

Answers:there really is nothing to explain - the formulas are in your book, just plug and grind and look up the answer in the normal distribution tables. 1. standard error of the mean = ( n ) = 0.75 / 700 = .002834733 2. 2 * P ( ( .085 ) ( n ) ) = 99.7% 3. P ( (8.185-8.1)/(.75/ 700) ) - P ( (8.157-8.1)/(.75/ 700) ) = 0.02081 = 2.1%