Find the time period of the oscillation of mass m in figure a,b,c what is the equivalent spring constant of the pair oif springs in each case?

Answer

`=2pi sqrt(m/(k_1+k_2))`
`=2pi (sqrt(m(k_1+k_2))/(k_1k_2))`

Explanation

`a`. Equivalent spring constant
`=k=k_1+k_2(para ll el)`

`T=2pisqrt(m/k)=2pisqrt(m/(k_1+k_2)`
`b`. Let us displace theh block m towards left through dislacement x.
Resultant force `F=F_1+F_2=(k_1+k_2)`
Acceleration
`(F/m)=((k_1+k_2))/mx`
`Time period`
`=T=2pisqrt(displacement/Acceleration)`
`rarr 2pisqrt(x/x((k_1+k_2)/m))`
`=2pi sqrt(m/(k_1+k_2))`
The equivalent spring constant
`k=k_1+k_2` . `c` In series connectionn equivalent spring constant be k
So,k `1/k=1/k_1+1/k_2=(k_2+k_1)/(k_1k_2)`
`rarr K=(k_1k_2)/(k_1+k_2)`
`T=2sqrt(m/k)`
`=2pi (sqrt(m(k_1+k_2))/(k_1k_2))`