Special Relativity - Length Contraction & Relativistic mass:

What is length contraction?

• The length of an object, moving at a constant speed

• .. with respect to an observer, appears to be shortened in the direction of its motion.

• i.e. looks longer when on the spaceship.

If an object is moving at a greater speed, what does this result in?

• Greater length contraction.

What is proper length? (Lo)

• Length measured by observer moving with the object.

What is L?

• Length measured by the stationary observer.

• Lo >L

What are the effects of relativistic mass?

• The faster an object moves (relative to an observer)..

• the more massive it gets

• …i.e. increase velocity, increase mass

What is mo?

• The smallest mass value

• .. when its at v = 0

• also known as rest mass

What is m?

• The largest mass value of the object.

When an object reaches the speed of light, what happens to its mass?

• No object can move with a velocity greater than the speed of light

• so when v = 0, its mass = infinity

Draw a graph of mass against velocity in terms of relativistic mass:

Draw a graph of total energy against speed for relativistic energy: Explain with some graphical analysis:

• As you increase velocity

• …(meaning work is being done on an object/energy is being supplied)

• The energy of the object becomes > its rest energy

• .. as its velocity goes to the speed of light

• … its energy = infinity.

When the velocity of an object is 0, what is its total energy?

• Its rest energy.

What is the rest energy calculation?

• Eo = moc²

How can you find the total energy of an object?

• Rest energy + Kinetic energy

How can you calculate the kinetic energy of an object moving at relativistic speeds (very large)?

• As you know that the total energy of the object = rest energy + kinetic energy

• .. you can rearrange this so Ek = Et - Eo

• where Et = Moc²/ root(1-v²/c²)

• & Eo = moc²

Why can’t you calculate the kinetic energy of an object travelling at relativistic speeds through 1/2mv²?

• Proven through Bertrozzi’s experiment

What does Bertrozzi’s experiment include?

• It involved a particle accelerator which could emit bunches of electrons

• at varying kinetic energies

• through two detectors plates

• connected to an oscilloscope

• and an aluminium target plate connected to a temperature sensor

Explain Bertrozzi's experiment:

1) The electrons (in bunches) were emitted in pulses.

2) The time taken for them to travel from the detector to the aluminium target disc could be calculated through the oscilloscope.

3) The distance between the detector and the aluminium target disc is also measured.

4) ..and the speed of the electrons was calculated (through v = s/t)

5) The electrons are directed at the aluminium target and when they collide with it, their kinetic energy is transferred to the target in the form of heat.

6) The change in temperature of the target is measured using the temperature sensor.

7) meaning the kinetic energy of the electrons could be directly measured.

How could the kinetic energy of one electron in the experiment be calculated?

• Energy = mc(change in temp)

• … which also will equal to the kinetic energy of all the electrons

• .. so you divide this by the number of electrons.

What are two precautions required for this experiment?

• That the distance of the wire from from the detector to the oscilloscope is THE SAME as the distance of the wire from the aluminium target disc to the oscilloscope

• ..to avoid a systematic timing error

• That no energy is lost in the process of kinetic energy transferring to heat

• .. this can be checked by measuring the heat transferred in the aluminium block

• …Q = mc x change in temp should also equal W = qVa..

• where q = total charge of the bunch of electrons.

What did Bertrozzi find from his experiment?

• He found that the maximum speed of these electrons were approximately the speed of light

• .. implying that when kinetic energy increased, so did mass.

• He also found out then when the accelerating voltage increased, so did work done

• .. but we don’t get an increase in velocity according to 1/2mv²

When plotting a kinetic energy graph against speed graph, what did the classical & relativistic plots show?

• The classical Ek points showed that the formula 1/2mv² cannot be applied to objects moving relativisticly ( very fast)

• ..as it implies that they can travel greater than the speed of light

• .. which is not true.