CBSE Class 11-science Answered

4th sum

Asked by lovemaan5500 | 04 Oct, 2018, 06:33: PM

Expert Answer

Let ΔT be the required increase in temperature so that heated Aluminium sphere will just pass through heated brass ring.

volume of sphere after heating

begin mathsize 12px style equals space 1 over 6 pi cross times d subscript s 0 end subscript superscript 3 cross times open parentheses 1 plus gamma increment T close parentheses space equals space 1 over 6 pi cross times d subscript s superscript 3 end style

.......................(1)

where d_s0 andd_s are initial and final diameter of sphere. γ is the volume expansion coefficient of Aluminium

Eqn.(1) is rewritten so that,

begin mathsize 12px style d subscript s space equals space d subscript s 0 end subscript open parentheses 1 plus gamma increment T close parentheses to the power of bevelled 1 third end exponent end style

................................(2)

circumfrence of ring after heating = π×d_r = π×d_r0×(1+αΔT) or d_r = d_r0×(1+αΔT) .................(3)

where d_r0 andd_r are initial and final diameter of sphere. α is the linear expansion coefficient of brass

diameters given by eqn.(3) and eqn.(2) are same if the sphere just pass through ring.

hence by equating (2) and (3), we get

begin mathsize 12px style fraction numerator open parentheses 1 plus gamma subscript s increment T close parentheses to the power of begin display style bevelled 1 third end style end exponent over denominator open parentheses 1 plus alpha subscript r increment T close parentheses end fraction space equals fraction numerator begin display style open parentheses 1 plus fraction numerator gamma subscript s increment T over denominator 3 end fraction close parentheses end style over denominator open parentheses 1 plus alpha subscript r increment T close parentheses end fraction space equals space fraction numerator begin display style open parentheses 1 plus alpha subscript s increment T close parentheses end style over denominator open parentheses 1 plus alpha subscript r increment T close parentheses end fraction space equals space d subscript r 0 end subscript over d subscript s 0 end subscript space equals space fraction numerator 24.9 over denominator 25 end fraction equals 0.996 end style

...............(4)

In eqn.(4), we used the approximation (1+x)ⁿ ≈ 1+nx, if x << 1.

Also α_s and α_r are expansion coefficients of sphere(Aluminium) and ring(brass) respectively

we used the relation γ = 3×α for expansion coefficients.

By substituting the values for expansion coefficient in eqn.(4), we can solve for ΔT to get ΔT = 500 ºC

Since initial temperature is 20ºC, Aluminum sphere will pass through brass ring at 520 ºC .

Answered by Thiyagarajan K | 04 Oct, 2018, 09:35: PM

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