IP It is desired to slip an aluminum ring over a steel bar (see the figure(Figur

Question

IP It is desired to slip an aluminum ring over a steel bar (see the figure(Figure 1)). At 10 degree C the inside diameter of the ring is 5.160 cm and the diameter of the rod is 5.200 cm. In order for the ring to slip over the bar, the ring is heated. Once the aluminum ring is slipped over the bar, the ring and bar are allowed to equilibrate at a temperature of 10 degree C. The ring is now stuck on the bar. Find the temperature at which the ring can be removed. Express your answer using two significant figures.

Explanation / Answer

Let at t*c the difference in dia of steel & aluminum is 0.
=>By A = Ao[1 + t]
=>d^2 = do^2[1+t]
=>d = do x [1+t]
=> for steel:-
=>d(s) = 5.200 x 10^-2 x [1+2 x 12 x 10^-6 t]------------(i) {as = 2}
=>For alumimum:-
=>d(a) = 5.160 x 10^-2 x [1+2 x 23.1 x 10^-6 t]------------(ii)
=> for removal of ring
=>ds = da
=>5.200 x 10^-2 x [1+2 x 12 x 10^-6 t] = 5.160 x 10^-2 x [1+2 x 23.1 x 10^-6 t]
=>5.200/5.160 = [{1+2 x 23.1 x 10^-6 t}/{1+2 x 12 x 10^-6 t}]
=>(1.0077)^2 = {1+2 x 23.1 x 10^-6 t}/{1+2 x 12 x 10^-6 t}
=>1.0155639683 x {1+2 x 12 x 10^-6 t} = {1+2 x 23.1 x 10^-6 t}
=>[2 x 23.1 x 10^-6 - 2 x 12 x 10^-6] x t = 1.0155639683 - 1
=> t = 701.08*C
=>t = 701.08 + 10 = 711.08*C

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