1 relativity h4: some consequences of special relativity

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RelativityRelativity

H4: Some consequences of special H4: Some consequences of special relativityrelativity

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Twin paradoxTwin paradox

An interesting and amusing result predicted An interesting and amusing result predicted by relativity theory is often called the twins by relativity theory is often called the twins "paradox". "paradox".

If one of a pair of twins goes on a long, fast If one of a pair of twins goes on a long, fast journey and then returns home, it will be journey and then returns home, it will be found that the twins have found that the twins have aged differentlyaged differently

the "stay-at-home" twin is always the older the "stay-at-home" twin is always the older

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Twin paradoxTwin paradox

The special theory of relativity deals only The special theory of relativity deals only with inertial frames of reference, the with inertial frames of reference, the astronaut twin would have undergone astronaut twin would have undergone huge accelerationshuge accelerations, thus his frame is , thus his frame is not inertialnot inertial

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Hafele-Keating Hafele-Keating experimentexperiment

This was an experiment to demonstration This was an experiment to demonstration the twin paradox.the twin paradox.

Two atomic clocks were flown around the Two atomic clocks were flown around the world in opposite directions.world in opposite directions.

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The clocks were compare with a The clocks were compare with a stationary clock.stationary clock.

Eastward Journey Westward Eastward Journey Westward JourneyJourney

Predicted Predicted -40 +/- 23 ns + 275 +/- 21 ns-40 +/- 23 ns + 275 +/- 21 ns

Measured -59 +/- 10 ns + 273 +/- 7 nsMeasured -59 +/- 10 ns + 273 +/- 7 ns

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This experiment showed that the clocks This experiment showed that the clocks had experienced time dilation, compare had experienced time dilation, compare to the stationary clock.to the stationary clock.

One clock had slowed down and one One clock had slowed down and one clock had sped upclock had sped up

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Vector additionVector addition

At normal speeds, At normal speeds, you just add the you just add the velocities, at velocities, at relativistic speeds relativistic speeds the correct formula isthe correct formula is

if ‘u’ is in the same if ‘u’ is in the same direction as ‘v’direction as ‘v’

2

'

'

1cvu

uvu

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Mass and energyMass and energy

The mass of a body is a The mass of a body is a relativerelative concept. concept. The mass of a body measured by an observer The mass of a body measured by an observer

at rest relative to the body is called (not at rest relative to the body is called (not surprisingly) the surprisingly) the rest rest mass of the body.mass of the body.

The mass (sometimes called the relativistic The mass (sometimes called the relativistic mass) of the body measured by other mass) of the body measured by other observers depends on the velocity of the observers depends on the velocity of the observer relative to the body.observer relative to the body.

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As the variation of mass As the variation of mass is basically due to the is basically due to the time dilation effect, you time dilation effect, you should not be surprised should not be surprised to find that, if the rest to find that, if the rest mass of a body is mo, mass of a body is mo, then its mass, m, as then its mass, m, as measured by an measured by an observer moving with observer moving with speed v relative to the speed v relative to the body is given bybody is given by

2

20

1cv

mm

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The Einstein relationships are:The Einstein relationships are: E = mcE = mc22 (total energy)(total energy) E = mE = m00cc22 (rest energy)(rest energy)

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If a force causes body If a force causes body BB to accelerate to accelerate away from observer away from observer AA (as in the example (as in the example above) then above) then work work is done by that force.is done by that force.

As usual we can define the kinetic energy As usual we can define the kinetic energy possessed by body possessed by body BB (as measured by (as measured by AA) by saying that ) by saying that

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K.E. of K.E. of BB = Work Done causing it to = Work Done causing it to accelerateaccelerate

It can be shown that if the relative speed It can be shown that if the relative speed of body of body BB is such that its mass (as is such that its mass (as measured by measured by AA) is m, then) is m, then

K.E. = mcK.E. = mc22 - m - moocc22 or K.E. = (change in or K.E. = (change in

mass)cmass)c22

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Here we are seeing the Here we are seeing the equivalenceequivalence of of mass and energy. Some of the work mass and energy. Some of the work done by the force is converted into mass done by the force is converted into mass and if we define the total energy, E, and if we define the total energy, E, possessed by a body to be the sum of its possessed by a body to be the sum of its rest energyrest energy (m (moocc22) and its K.E. we have ) and its K.E. we have

the famous resultthe famous result

E = E = mcmc22

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It is easy to show that if the velocity of the body It is easy to show that if the velocity of the body relative to the observer is small relative to the observer is small compared with compared with the velocity of lightthe velocity of light, then the relativistic formula , then the relativistic formula reduces to the Newtonian expression (½mv2)reduces to the Newtonian expression (½mv2)

If the velocity of the body relative to the If the velocity of the body relative to the observer is very close to the velocity of light, observer is very close to the velocity of light, virtually virtually all the work done by the force is all the work done by the force is converted to massconverted to mass..

Experiments involving high speed particles Experiments involving high speed particles (protons, electrons etc.) in particle accelerators (protons, electrons etc.) in particle accelerators give evidence to support the idea that mass give evidence to support the idea that mass varies with velocity. varies with velocity.

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Units of Mass/EnergyUnits of Mass/Energy

The S.I. unit for mass is the kg and for energy, The S.I. unit for mass is the kg and for energy, the Joule. However, on the scale of subatomic the Joule. However, on the scale of subatomic particles, we often use MeV for energy (1MeV particles, we often use MeV for energy (1MeV = 1·6×10= 1·6×10-13-13J). So, a possible unit for the J). So, a possible unit for the quantity "mcquantity "mc22" is MeV. " is MeV.

For this reason, the masses of subatomic For this reason, the masses of subatomic particles are often expressed in MeV/cparticles are often expressed in MeV/c22..

For example the rest energy of an electron is For example the rest energy of an electron is about 511MeV and its mass is therefore said to about 511MeV and its mass is therefore said to be 511MeV/cbe 511MeV/c22..

1 relativity h4: some consequences of special relativity

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