Find the elastic potential energy stored in each spring shown in figure, when the block is in equilibrium. Also find the time period of vertical oscillation of the block.

Answer

`=2pi sqrt(M(1/k_1+1/k_2+1/k_3))`
`PE_1=(m^2g^2)/(2k_1)`
`PE_2=(m^2g^2)/(2k_2)`
`PE_3=(m^2g^2)/(2k_3)`

Explanation

`k_1, k_2, k_3` are in series
`=1/k=1/k_1+1/k_2+1/k_3`
`rarr K=(k_1k_2k_3)/(k_1k_2+k_2k_3+k_3k_1)`
time period
`T=2pisqrt(m/k)`
`=2pi sqrt(M(1/k_1+1/k_2+1/k_3))`
Now force =weight =mg
:. At `k_1` spring
`x_1=(mg)/k_1`
similarly `x_2=(mg)/k_2`
and `x_3=(mg)/k_3`
`:. PE_1=1/2k_1x_1^2`
`=1/2k_1((Mg)/K_1)^2`
`=1/2k_1(m^2g^2)/k_1^2`
`=1/2 (m^2g^2)/k_1=(m^2g^2)/(2k_1)`
similarly `PE_2=(m^2g^2)/(2k_2)`
and `PE_3=(m^2g^2)/(2k_3)`