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Find the centre of mass of a uniform pla...

Find the centre of mass of a uniform plate having semicircular inner and outer boundaries of radii `R_1 and R_2`.

Text Solution

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`y_(cm)=(intydm)/M`
or `dm=(Mpirdr)/(pi(R_2^2-R_1^2)/2`
`M/(R_2^2-R_1^2)/2`

`y_(CM)=int_R_1^R_2((2r)/pi).M/(R_2^2-R_1^2)/2.(rdr)/M`
`=2/pi (R_2^2-R_1^2)/2 int_R_1^R_2r^2 dr`
`=2/pi(R_2^2-R_1^2)/2[(R_2^3-R_1^3)/3]`
`=(4(R_1^2+R_1R_2+R_2^2)/(3pi(R_1+R_2)`
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Knowledge Check

  • The moment of inertia of a uniform semicircular disc of mass disc through the centre is

    A
    `(2)/(5)Mr^2`
    B
    `(1)/(4)Mr^2`
    C
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    D
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  • Find the coordinates of centre of mass of a uniform semicircular closed wire frame with respect to the origin which is at its centre. Radius of the circular portion is R

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    C
    `((2R)/(pi),0)`
    D
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  • In the figure shown, find out centre of mass of a system of a uniform circular plate of radius 3R from O in which a hole of radius R is cut whose centre is at 2R distance from the centre of large circular plate

    A
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    B
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    C
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    D
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