A body executes S.H.M under influence of a force with a time period of 1.0s. It has a time period of 1.5s under theaction of a force of different magntitude. What will be the period of oscillation of the body when these two force are impressed simultaneously in the same direction upon the same body?

Explanation

We know that `Tprop (1)/(sqrta)` where T - period of oscillation and a = acceleration
Also `F=ma therefore a prop " Force"`
`therefore prop (1)/(sqrtF)," "T_(1)prop (1)/(F_(1))" ………….(1)"`
`T_(2)prop (1)/(sqrt(F_(2)))" ............(2)"`
i.e. `(T_(1))/(T_(2))=sqrt((F_(2))/(F_(1)))" i.e. "((1)/(1.5))^(2)=(F_(2))/(F_(1))`
`"or "(F_(1))/(F_(2))=2.55" "F_(1)=2.25F_(2)`
When the combined forces act on the body then,
`T prop (1)/(sqrt(F_(1)+F_(2)))" .........(4)"`
`(4)+(1)" gives "(T)/(T_(1))=sqrt((F_(1))/(F_(1)+F_(2)))`
i.e. `(T)/(T_(1))=sqrt((F_(1))/(F_(1)(1+(F_(2))/(F_(1)))))=sqrt((1)/(1+((F_(2))/(F_(1)))))=sqrt((1)/(1+((1)/(2.25))))=sqrt((1)/(1.444))`
`therefore" "T=0.8322T_(1)`
`"i.e. "T=0.8322xx1s=0.8322s.`