Question

A pendulum clock is 5sec fast at temperature of 15oc and 10sec slow at a temperature of 30oc. At temperature does it give the correct time-

Answer 1

A simple method to solve this is the following:Time lost or gained= aplha∆t/2 X given time periodCase 1: Time gained is 5s so the condition is like this: 5=alpha(t-15)/2 where it`s (t-15) as it`s gained so t is at more temp.Case 2: Time lost is 10s so the condition is like this: 10=alpha(30-t)/2 where it`s (30-t) as it`s lost so t is at a less temp.Dividing these two equations we get the temp at which clock gives correct time.10/5=(30-t)/2 X 2/(t-15)2(t-15)=30-t2t-30=30-t3t=60T=20°

FORMULA:-We know that: T = 2π √(l/g)

dT/T= 1/2∝ ∆T

And we can say that temp., would be between 15oC and 30oC 

5/T=1/2∝(θ- 15) ------1nd

10/T= 1/2  ∝(30- θ)  ------2nd

Diviving 1st and 2nd eqn

we get, 

5/10= (θ-15)/(30- θ)  

⸫ θ = 20oC