college physic

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Determine a formula for the change in surface area of a uniform solid sphere of radius r if its coefficient of linear expansion is alpha A (assume constant) and its temperature is changed by Delta T.Assume the temperature chan ge is not large, Delta T^2 ~ 0.

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    Area A = 4 pi R^2
    lnA = ln(4 pi) + 2 ln R
    dA/A = 2 dR/R = 2*(alpha)*dT
    delta A = 2A*alpha*(delta T)


    A at T + deltaT
    delta A = 4 pi (R + deltaR)^2 - A
    = 4 pi R(1 + alpha*deltaT)^2 -A
    = 4 pi R^2 *[1 + 2*alpha*deltaT +
    (alpha*deltaT)^2] - A
    = 2*A*alpha*deltaT + A*(alpha*deltaT)^2

    (Ignore the second term in deltaT^2)

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