4.3 Solutions
155
4.7 The moment of inertia of the plate about x-axis is I x = (1/3) Mb 2 and about
y-axis is I y = (1/3) Ma 2 . It can be shown that the moment of inertia about the
line BD is
I BD =
1
3
Mb
2 cos
2
θ +
1
3
Ma
2 sin
2
θ
(1)
where θ is the angle made by BD with the x-axis. From Fig. 4.20, cos θ =
a
√
a 2 + b 2
and sin θ =
b
√
a 2 + b 2
.
∴ I BD =
1
3
M b 2 a 2
a 2 + b 2
+
1
3
M a 2 b 2
a 2 + b 2
=
2
3
M
a 2 b 2
a 2 + b 2
Fig. 4.20
4.8 The moment of inertia about any side of a triangle is given by the product of
the one-sixth mass m of the triangle and the square of the distance ( p) from the
opposite vertex, i.e. I = mp 2 /6. The perpendicular BD on AC is found to be
equal to 12/5 from the geometry of Fig. 4.21.
Fig. 4.21
155
4.7 The moment of inertia of the plate about x-axis is I x = (1/3) Mb 2 and about
y-axis is I y = (1/3) Ma 2 . It can be shown that the moment of inertia about the
line BD is
I BD =
1
3
Mb
2 cos
2
θ +
1
3
Ma
2 sin
2
θ
(1)
where θ is the angle made by BD with the x-axis. From Fig. 4.20, cos θ =
a
√
a 2 + b 2
and sin θ =
b
√
a 2 + b 2
.
∴ I BD =
1
3
M b 2 a 2
a 2 + b 2
+
1
3
M a 2 b 2
a 2 + b 2
=
2
3
M
a 2 b 2
a 2 + b 2
Fig. 4.20
4.8 The moment of inertia about any side of a triangle is given by the product of
the one-sixth mass m of the triangle and the square of the distance ( p) from the
opposite vertex, i.e. I = mp 2 /6. The perpendicular BD on AC is found to be
equal to 12/5 from the geometry of Fig. 4.21.
Fig. 4.21