Question

find the centre of mass of a rectangular lamina of length l and width b​

Answer 1

Answer:

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Answer 2

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

)

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