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Locate c.m. of thin , uniform semicircul...

Locate `c.m.` of thin , uniform semicircular wire of radius `R`.

Text Solution

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`dm = (m)/(pi R) R d theta = (m d theta)/(pi)`
`y_(c.m.) = (int y dm)/(int dm) = (int_(0)^(pi) R sin theta (m)/(pi) d theta)/(m) = (R )/(pi) int_(0)^(pi) sin theta d theta`
`= (R )/(pi) | - cos theta|_(0)^(2) = (R )/(pi) [ - cos (pi) - { - cos(0)}]`
`=(R)/(pi) [ 1 + 1] = ( 2R)/(pi)`
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Knowledge Check

  • Consider following statements [1] CM of a uniform semicircular disc of radius R=2R//pi from the centre [2] CM of a uniform semicircular ring of radius R=4R//pi from the centre [3] CM of a solid hemisphere of radius R=4R//3pi from the centre [4] CM of hemisphere shell of radius R=R//2 from the centre Which statements are correct

    A
    1,2,4
    B
    1,2,3
    C
    4 only
    D
    1,2only

  • Consider following statements (1) CM of a uniform semicircular disc of radius R is 2R/ pi from the centre (2) CM of a uniform semicircular ring of radius R is 4R/3 pi from the centre (3) CM of a solid hemisphere of radius R is 4R/3 pi from the centre (4) CM of a hemisphere shell of radius R is R/2 from the centre Which statements are correct?

    A
    1, 2, 4
    B
    1, 3, 4
    C
    4 only
    D
    1, 2 only

  • The M.I. of a uniform semicircular disc of mass M and radius R about a line perpendicular to the plane of the disc and passing through the centre is

    A
    `(1)/(2)MR^(2)`
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    C
    `MR^(2)`
    D
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