IIT JEE , AIEEE Forum - Physics , Mechanics

coffeeadicto

find the distance of centre of mass of a uniform plate having semicircular inner and outer boundaries of radii a and b.
the ans is 4/3*

(a²+ab+b²)/(a+b)
plz tell me how to arrive at the soln.

hit_ur_heart

dude.. don't panic...here is the solution

First of all you need to calculate the COM of a thin semi circular ring of radius "r".
that part I leave it to you for practice.answer is "2r/

"

now in the disc consider a ring element of radius "x" and thickness "dx" where "x" varies from "a" to "b".
NOW remeber the formula for COM :
y_{com =_{[[ a ]}^{[b ]} (2x/) * ( x dx) * ] / M

where = surface mass density = 2M/}_{(b^2 - a^2)
You can check that out

...Now solve the integration and here comes ur answer}

edison

Good effort and approach suggested by hit_ur_...

I have used alternative approach as follows:

Here outer radius is = b

inner radius = a

If we consider

a) semicircular plate P₁ of radius 'b'.

b) Semicircular plate P₂ of radius 'a'.

c) & Given semicircular plate P₃ with inner and outer radii a and b respectively.

We intend to find CM for plate P₃.

Moreover by CM here i mean point on y-axis only as by symmetry of the problem x-ordinate is zero. (Origin is center of the plate and x and y axes are mutually perpendicular radii)

Now CM for P₁ = Y₁ = 2b/

CM for P₂ = Y₂ = 2a/

Let CM for P₃ = Y₃

Since, P₂ and P₃ are parts of P₁, thus

Y₁ = (M₂Y₂+ M₃Y₃)/(M₂ + M₃) ----(1)

where M₁ and M₂ are masses of plates P₁ and P₂

If density of the material of plate is 'd' then

M₂ =

a²d/2, M₃ =

d(b²- a²)/2

so from eq. (1)

Y₃ = [Y₁(M₂ + M₃) - M₂Y₂]/M₃

Substitute the values and find the CM for P₃ which is the desired result.

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