A compensated pendulum shown if Fig. is in the from of an isosceles Delta of base length `l_1=5`cm and coefficent of linar expansion `alpha_1=18xx10^-6` and side length `l_2` and coefficient of linear expansion `alpha_2=12xx10^-6`. find `l_2` so the the distance of centre of mass of the bob from suspension centre O may remain the same at all the temperature.

Explanation

First we muist know whatis a compensated pedulum. We've discussed that due to change in temperature thetime period of a pendulum clock changes. Due to this in pendulum clocks, to make a pendulum some specific emtals are used which have very low coefficient of expansion so that the error introduced in their time is very small. The other alternative of minimizing the error in time measurement, is to use compensated pendulum. This is a pendulum made up of two or more metals of such an arrangement in which the effective length of centre of masss of the whole body remains unchanged. Such a situation is given in figure.
In this case the distance of centre of mass from suspension point O is
`h=sqrt(l_(2)^(2)-(l_(1)^(2))/4)`
At some higher temperature `Deltat`, new value of h will become
`h'=sqrt(l_(1)^(2)(1+alpha_(2)Deltat^(2))+(l_(1)^(2))/4(1+alpha_(1)Deltat)^(2))`
As we require `h=h'` we have
`l_(2)^(2)=(l_(1)^(2))/4=l_(2)^(2)(1+2alpha_(2)Deltat)-(l_(1)^(2))/4(1+2alpha-(1)Deltat)`
or `2alpha_(2)l_(2)^(2)=(l_(1)^(2))/4alpha_(1)`
or `l_(2)=(l_(1))/2 sqrt((alpha_(1))/(alpha_(2)))`