- #1
Vibhor
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Homework Statement
Q. What is the time lost in time 't' by a pendulum clock whose actual time period is T and the changed time period at some higher temperature is T' ? (T' is Time period at some higher temperature , l' is the increased length , Δθ is the difference in temperature, α is coefficient of linear expansion)
Homework Equations
T = ##2\pi \sqrt{\frac{l}{g}}##
The Attempt at a Solution
Time period of a simple pendulum is given by T = ##2\pi \sqrt{\frac{l}{g}}## . As the tempertaure is increased , length of the pendulum and hence time period gets increased . the pendulum clock becomes slow and hence loses time .
If T' is Time period at some higher temperature , l' is the increased length , Δθ is the difference in temperature, α is coefficient of linear expansion .
##\frac{T'}{T} = \sqrt{\frac{l'}{l}}## = ## \sqrt{\frac{l+Δl}{l}}## = ## \sqrt{\frac{l+lαΔθ}{l}}## = ##(1+αΔθ)^{\frac{1}{2}}##
##T' ≈ T(1+\frac{1}{2} αΔθ)##
##ΔT = T'-T = \frac{1}{2} αΔθ##
Time lost/sec = ##\frac{ΔT}{T}##
Time lost in time 't' = ##\frac{ΔT}{T}t##
But the answer given is ##\frac{ΔT}{T'}t## . This is what I do not understand . When we calculate Time lost/sec , we should divide ΔT by T or ##T'## ( new/increased time period ) . I think it should be T , but according to the book it should be ##T'##
Please help me with the concept .
Many Thanks
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