Undergraduate Lecture Notes in Physics Gabor Kunstatter Saurya Das First Course on Symmetry Special Relativity and Quantum Mechanics the Foundations of Physics 1st Edition Instructor Sol

Undergraduate Lecture Notes in Physics Gabor Kunstatter Saurya Das First Course on Symmetry Special Relativity and Quantum Mechanics the Foundations of Physics 1st Edition Instructor Sol
Chapter 16
Solutions to Exercises
16.1 Introduction
No exercises
16.2 Symmetry and Physics
Exercise 1. Calculate the length D of roadway in Fig. 2.3 as a function
of l and a, and verify that it is minimized by the value of l given in Eq.(2.1)
above.
Solution:
Each diagonal segment on the right hand side of Fig. 2.3 has length:
d(l) = s
a
2
2
+

a − l
2
2
(16.1)
The total length of pavement D(l) as a function of l is:
D(l) = l + 4d = l + 4s
a
2
2
+

a − l
2
2
(16.2)
496
CHAPTER 16. SOLUTIONS TO EXERCISES 497
To find the value of lmin that minimizes the function D(l) we have to solve:
dD(l)
dl
 
 
 
 
lmin
= 1 +
2(a − l)(−1)
p
a
2 + (l − a)
2
= 0 (16.3)
A bit of algebra yields:
(a − lmin)
2 =
a
2
3
(16.4)
Since l must be less than a the relevant solution is:
lmin = a

1 −
1
√
3

(16.5)
Plugging this into the expression for D(l) we find:
D(lmin) = a

1 +
1
√
3

(16.6)
which is less than the length of pavement required by building two diagonal
roads directly through the center of the square, namely:
Ddiagonals = 2√
a
2 + a
2 = 2√
2a (16.7)
A straightforward calculation verifies that
d
2D(l)
dl2
 
 
 
 
lmin
< 0 (16.8)
so that lmin is indeed a minimum as required.
Exercise 2. Consider three towns, called N (for North), SW and SE,
respectively, located a distance a apart at the vertices of an equilateral triangle, as shown in Fig.[2.4]. We wish to build a network of roads connecting
all three towns such that the roads (shown in blue) meet at an arbitrary
point P along the line joining the center of the triangle to the northern town
at some distance l from N. Show that the minimum total length for such aconfiguration of roads occurs when l = a/√
3, so that the least expensive way
to join the towns is to have the three segments of road meet at the center,
C. Is there symmetry breaking in this case? Explain.
Hint: Use the law of cosines to figure out the distance from P to the other
two towns SW and SE. This should give you an expression for the total
length of the three roads that join P to N, SW and SE. Finally, minimize
the total length of pavement with respect to the parameter l and show that
the minimum occurs at l = a/√