You may pick up your R1 and R2 problems on the front desk. There are 5 points possible on each. The solutions are on-line, password is PH365.

R3 problems and the meter lab should be turned in now in the box on the front desk. Problems will be returned tomorrow

Problems that should be considered part of the practice test (You will see one of each type on the test) R4B.4 and R5B.6

Tomorrow will be a review day. When you come to class up to 15 points will be given for the material completed on the practice test.

Friday – test over relativity. Students who wish more time may start at 7:00 or remain until 10:00.

Chapter R5

Proper time

Calculating the path length from A to B – normal (Euclidian) space

y

x

dx

dy

Each small segment of path length is given by

dLdL2=dx2+dy2

A

B

Distance along the path If we sum all the segments we get the total

length.

B

A

B

A

B

A

AB dydy

dxdydxdLL 2

222 1(

Calculating the path length from A to B in a space time diagram

t

x

dx

dt

Each small segment of path length is given by

dτdτ2≈Δs2=dt2-dx2

A

B

Distance along the path If we sum all the segments we get the

total length.

B

A

B

A

B

A

AB dtdt

dxdxdtd 2

222 1)(

B

A

AB dtv21

In the case where v is a constant (speed is constant, not the velocity).

)(11 22AB

B

A

AB ttvdtv

Different times again We are measuring here the proper time

(τ) Same clock present at both events The clock may or may not be inertial If it is inertial it gives the spacetime interval

Coordinate time is the time measured by two synchronized clocks. Synchronized means they measure the

speed of light to be 3 x 108 m/s.

Example R5.1 A clock is tied on the end of a

string and swung in a 3m circle. How fast must the clock move around the circle to have a .01% difference in the times?

Event A – clock passes person holding standard clock.

Event B – clock passes the same place one turn later.

Both clocks measure proper time. One in the home frame and in the other frame

The speed is constant, but not the velocity. The moving clock is not inertial.

v=? v=0.014 v= 4200 km/s Much too fast to be realistic.

ABABAB tttv 9999.0)(1 2

Relationships between the times

Δt ≥ Δs ≥ Δτ The proper time is the shortest and

the coordinate time is the longest. The three are equal when both of

the events occur in the same place. The spacetime interval is the

longest possible proper time.

Another muon experiment (muons made in an accelerator) The half life of munons (a negative lepton with a

mass 200 times greater than the electron) is 1.52μs

A bunch of munons is injected into a storage ring with v=0.99942. How long does their half-life appear to be in the lab?

What kind of time is the half life? Coordinate, proper or spacetime interval?

Coordinate – the same clock is present at both events.

What kind of time is the half life as seen in the lab? Coordinate, proper or spacetime interval?

Proper (depends on the path which in this case is accelerated)

21

1

v

t

AB

AB

We want to calculate the ratio between ΔτAB and ΔtAB to see how many times longer the muons will live in the accelerator than if they were at rest.

AB

ABt

Calculate this for v=0.99942

An experiment done in 1977 by Bailey et al., Nature, volume 268, found this to be 29.327±0.004

In the storage ring the particles are going in a circle of radius 7.01 m with an acceleration of 1015 times that of gravity.

Definition

This expression comes up often so we make the following definition.

2

2

1c

v

2

2

1

1

c

Where β is the velocity of the other system with respect to the home system.

Calculate γ when β = 0.95c

γ = 3.2 Make a note of this number, you will use it later today.

Results we would have obtained if we had continued the study of relativity

Length L=L'/γ The home system measures the

length in the other system shorter than they do.

Mass m=γm' We measure the mass in the other

system to be greater than they do.

The best known result E = mc2

Where m is the rest mass. Mass can be changed to energy and

energy to mass. Both happen. When energy is changed to matter,

equal amounts of matter and anti-matter are formed.

When matter hits anti-matter, both are completely changed to energy.

The cosmic speed limit

When energy is added to an object traveling close to the speed of light, the mass increases faster than the velocity.

See example R10.3 for a perfect (matter-antimatter) rocket problem.

An even bigger problem if it were possible for anything to travel faster than the speed of light. On page 137, an example is worked

showing that if a faster than light signal were possible, causality would be violated. This means it would be possible to find a

coordinate system in which the TV would turn on before you pushed the on button!

In all coordinate systems, causally connected events must happen in the same order.

Example: A space capsule of mass 10,000 kg and length 10 m goes by us at a speed of .95c emitting a flash of light every 20 seconds by their clock. What do we measure the mass of the

capsule to be? Length? How often is the light flashing? In a fusion rocket, 1 kg of hydrogen

becomes 993 g of helium. How much hydrogen fuel must be burned to get the capsule moving 1000 m/s starting from rest? Consider only the energy, not the momentum in answering this question.

32,000 kg

3 m

Every 64 seconds

You may pick up your R1 and R2 problems on the front desk. There are 5 points possible on each. The solutions are on-line, password is PH365.

R3 problems and the meter lab should be turned in now in the box on the front desk. Problems will be returned tomorrow

Problems that should be considered part of the practice test (You will see one of each type on the test) R4B.4 and R5B.6

Tomorrow will be a review day. When you come to class up to 15 points will be given for the material completed on the practice test.

Friday – test over relativity. Students who wish more time may start at 7:00 or remain until 10:00.