Two simple pendulums with heavy bobs, one using iron wire and the other aluminium wire, are excited simultaneously. When the first completes 1000 oscillations, the other completes 1001 . When the temp

Question

Two simple pendulums with heavy bobs, one using iron wire and the other aluminium wire, are excited simultaneously. When the first completes 1000 oscillations, the other completes 1001 . When the temperature is raised by t ∘ C , they oscillate together. If their coefficients of linear expansion are 10 × 1 0 − 6 ∘ C − 1 and 30 × 1 0 − 6 ∘ C − 1 respectively, t is:

TestHub · Accepted Answer

The initial periods satisfy 1000 T i ​ = 1001 T a ​ Equal final periods require α i ​ = 1 0 − 5 , α a ​ = 3 × 1 0 − 5 r = 1000 1001 ​ r 1 + α i ​ t ​ = 1 + α a ​ t ​ r 2 ( 1 + α i ​ t ) = 1 + α a ​ t ( α a ​ − r 2 α i ​ ) t = r 2 − 1 t = α a ​ − r 2 α i ​ r 2 − 1 ​ t ≈ 100.15 ∘ C

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