Concepts in Physics – Dr. P.A. Mulheran – January 2002
Week 10 Workshop Questions
Hand your solutions to the following problems to Dr Mulheran at the start of the next
workshop. Some of your work will be marked and will form part of the continuous assessment
element of this course. Late scripts will not be accepted except in extenuating circumstances
that are confirmed by your tutor. Illegible work will not be marked; rewrite your solutions if
necessary. You must also explain the reasoning of your work.
1. An atomic clock is raised to a height of 10km above the Earth's surface in a weather
balloon. Show that General Relativity predicts that this clock gains 4 ns per hour over a
similar clock that remains at sea-level. Is this difference measurable?
[5 marks]
2. (a) A clock moves from x=0 to x=L in time T as measured in the rest frame of a
laboratory. If the clock moves in a straight line at a constant speed v = L/T what time does
it record for its transit?
[2 marks]
(b) If the clock deviates from the straight line, but still moves at a constant speed so that it
arrives at x=L at time T in the laboratory frame, how does its recorded time of transit
change?
[2 marks]
(c) What is the relationship between Newton's 1st Law of Motion and Einstein's Law of
Motion?
[1 mark]
3. (a) Given that the frequency of a clock at gravitational potential  is increased by the
factor
 

1  c 2 
compared to a reference clock, what is the excess rate at the surface of the Earth due to
the Earth's gravitational mass?
[1 mark]
(b) The excess speed of clocks can be equated to the excess length of rulers as in Special
Relativity, where the time dilation factor is the same as the length contraction factor.
Hence estimate the excess radius of the Earth due to its gravitational mass (as might be
measured from the surface area of a sphere surrounding it).
[2 marks]
(c) The excess radius referred to in your lecture notes was derived from comparing the
radius measured through the centre of the Earth with that derived from its surface area.
Calculate an expression for excess radius in this version of the argument, showing why it
differs by a factor of one third from your answer in part (b).
[2 marks]
1
Concepts in Physics – Dr. P.A. Mulheran – January 2002
WORKSHOP SOLUTIONS
1. Rate of clock is
   gH 

2
4
17
2 2
12
1  c 2   1  c 2   1  10ms .10 m.10 m s  1  10
Thus in one hour = 3600 s the clock gains ~10-12 x 4000 s = 4ns as required.
Atomic clocks are based upon the magnetic dipole transitions of Cs atoms, which have a
sharp resonance at about 1010 Hz. This allows the construction of electronic oscillator circuits
with this (or in fact better) clock speeds, so ns precision is easily achieved. See Physics
World, Jan 2001 for more details.
[5 marks]
T
as in Special Relativity discussed in the previous lectures (recall moving
 v 
clocks run slow and  > 1).
[2 marks]
(b) The path length will be longer if the moving clock deviates from the straight line,
hence its speed must increase. This makes  larger and T' smaller.
[2 marks]
2. (a) T  
T
on the moving clock occurs for the straight line path with
 v 
constant speed. Thus Newton's 1st Law of Motion, that bodies move at constant speed
unless acted on by a force, follows naturally from Einstein's law that clocks move to
maximise their 'proper' time.
[1 mark]
(c) Maximum time T  
   GM E   gR E 

3. (a) 1  2   1 
 1  2   1  7.10 10 ,
2 
RE c  
c 
 c  
-2
6
g = 9.81ms , RE = 6.4x10 m, c = 3x108ms-2.


[1 mark]
(b) Excess radius = 7.10-10xRE = 45x10-4m = 4.5mm
[2 marks]
(c) Integrate step lengths along radius of Earth, taking into account the varying
gravitational potential with position (see workshop questions in week 2):
RE 
 1 GM E 
GM r 2 
Measured radius RM   1  2 E 3 .dr  RE 1 
.
2 
0
c RE 

 3 RE c 
[2 marks]
2