Moment of inertia of rectangular plate about an axis passing through p and
Answers
Answer:
Let the mass density be ρkg/m
2
Let the side PQ=RS=a & QR=PS=b (From fig a>b)
Mass of rectangle is m= ρab
Therefore Moment of Inertia of Rectangle about its center = m
12
a
2
+b
2
Distance of P point from center of rectangle is
2
a
2
+b
2
Therefore Moment of Inertia of Rectangle about P, I= m
12
a
2
+b
2
+m
4
a
2
+b
2
=m
3
a
2
+b
2
Mass of triangle PQR=
2
m
=
2
ρab
Moment of Inertia of Triangle PQR about its centroid = ρ
12
ab
3
+ba
3
=m
12
a
2
+b
2
Distance of point P from centroid =
(
3
2a
)
2
+(
3
b
)
2
Moment of Inertia of Triangle PQR about P= m
12
a
2
+b
2
+
2
m
{(
3
2a
)
2
+(
3
b
)
2
}=
2
m
18
11a
2
+5b
2
>
2
m
18
6a
2
+6b
2
>
2
I
(As a>b)
Distance of point R from centroid =
(
3
2b
)
2
+(
3
a
)
2
Moment of Inertia of TrianglePQR about P= m
12
a
2
+b
2
+
2
m
{(
3
a
)
2
+(
3
2b
)
2
}=
2
m
18
5a
2
+11b
2
As a>b therefore
2
m
18
5a
2
+11b
2
can be less than or equal to or greater than
2
I