Page 189 - Intro to Tensor Calculus

184

Change the given equations from a tensor notation to a vector notation.

i
I 33. ijk B k,j + F =0
I 34. g ij jkl B l,k + F i =0
∂%
I 35. +(%v i ),i =0
∂t
2
∂v i ∂v i ∂P ∂ v i
I 36. %( + v m )= − + µ + F i
m
∂t ∂x m ∂x i ∂x ∂x m
ZZ
I 37. The moment of inertia of an area or second moment of area is defined by I ij = (y m y m δ ij −y i y j ) dA
A
where dA is an element of area. Calculate the moment of inertia I ij ,i, j =1, 2 for the triangle illustrated in
1 bh 3 1 b h
2 2

the figure 2.1-1 and show that I ij = 12 2 2 − 24 3 .
1
1
− b h b h
24 12

Figure 2.1-1 Moments of inertia for a triangle

I 38. Use the results from problem 37 and rotate the axes in figure 2.1-1 through an angle θ to a barred
system of coordinates.
(a) Show that in the barred system of coordinates

I 11 + I 22 I 11 − I 22
I 11 = + cos 2θ + I 12 sin 2θ
2 2

I 11 − I 22
I 12 = I 21 = − sin 2θ + I 12 cos 2θ
2

I 11 + I 22 I 11 − I 22
I 22 = − cos 2θ − I 12 sin 2θ
2 2
(b) For what value of θ will I 11 have a maximum value?
(c) Show that when I 11 is a maximum, we will have I 22 a minimum and I 12 = I 21 =0.