find the centre of mass of a rectangular lamina of length l and width b
Answers
ANSWER
The moment of inertia about a rectangular lamina with area density ρ is ∫∫ρr
2
dA. Here r is the distance from each point to the axis of rotation and dA is the differential of area.
Take the center of the rectangle to be at (0,0) so the vertices are (L/2,B/2), etc. Then the distance from (x,y) to the axis of rotation (passing through (0,0) perpendicular to the plate) is x
2
+y
2
and the moment of inertia is
∫
−L/2
L/2
∫
B/2
B/2
ρ(x
2
+y
2
)dxdy
or
∫
−L/2
L/2
ρ(Bx
2
+
12
1
B
3
)
or
12
1
ρ(L
3
B+LB
3
)
Now as M=ρLB, thus we get moment of inertia as
12
M
(L
2
+B
2
)
so radius of gyration is K=
M
I
⟹K=
12
1
(L
2
+B
2
)
hope it help you plzz. mark as brainlist
Answer:
ANSWER
The moment of inertia about a rectangular lamina with area density ρ is ∫∫ρr
2
dA. Here r is the distance from each point to the axis of rotation and dA is the differential of area.
Take the center of the rectangle to be at (0,0) so the vertices are (L/2,B/2), etc. Then the distance from (x,y) to the axis of rotation (passing through (0,0) perpendicular to the plate) is x
2
+y
2
and the moment of inertia is
∫
−L/2
L/2
∫
B/2
B/2
ρ(x
2
+y
2
)dxdy
or
∫
−L/2
L/2
ρ(Bx
2
+
12
1
B
3
)
or
12
1
ρ(L
3
B+LB
3
)
Now as M=ρLB, thus we get moment of inertia as
12
M
(L
2
+B
2
)
so radius of gyration is K=
M
I
⟹K=
12
1
(L
2
+B
2
)
hope it help you plzz. mark as brainlist