Question

A body executes SHM under the influence of one force and has a period $T_1$ seconds and the same body executes SHM with period $T_2$ second when under the influence of another force. When both forces act simultaneously and in the same direction then the time period of the same body is:

• A. $(T_1+T_2)sec$
• B. $\sqrt{T_1^2+T_2^2}$
• C. $\sqrt{\frac{T_1^2+T_2^2}{T_1{T}_2}}sec$
• D. $\sqrt{\frac{T_1^2{T}_2^2}{(T_1^2+T_2^2)}}sec$

Answer 1

The correct option is D $\sqrt{\frac{T_1^2{T}_2^2}{(T_1^2+T_2^2)}}sec$

$\text{Given:-}$ Time oscillation of the body = $T_1$

Time oscillation of the body = $T_2$

$\text{To find:-}$ The time period of the same body

$\text{Solution:-}$

Under the force $F_1$ time oscillation of the body $T_1$

$F_1=M{\left(\frac{2\pi }{T_1}\right)}^{2}V$

where M is th emass and v is the velocity

$and$

$F_2=M{\left(\frac{2\pi }{T_2}\right)}^{2}V$

$F_{Net}=F_1+F2$

$M{\left(\frac{2\pi }{T}\right)}^{2}V=M{\left(\frac{2\pi }{T_1}\right)}^{2}V+M{\left(\frac{2\pi }{T_2}\right)}^{2}V$

$\frac{1}{T^2}=\frac{1}{T_1^2}+\frac{1}{T_2^2}$

$\frac{1}{T^2}=\frac{T_1^2+T_2^2}{T_1^2\times T_2^2}$

$\frac{1}{T}=\sqrt{\frac{T_1^2{T}_2^2}{T_1^2+T_2^2}}sec$

$\text{Hence the correct option is D}$