The springs shown in the figure are all unstretched in the beginning when a man starts pulling the block. The man exerts a constant force F on the blcok. Find the amplitude and the frequency of the motion of the block.

Answer

`T=2pisqrt(M/k)`
`=2pi sqrt((m(k_2+k_3))/(k_2k_3+k_1k_2+k_1k_3))`
Frequency `=1/T`
`=1/(2pi)sqrt((k_2+k_3+k_1k_2+k_1k_3)/(M(k_2+k_3)))`
Amplitude =`x
=`F/k`=(F(k_2+k_3))/(k_1k_2+k_2k-3+k_1k_3)`

Explanation

`k_2 and k_3` are in series.
let equivalent spring const. be `k_4`
`:. 1/k_4=1/k_2+1/k_3=(k_2+k_3)/(k_2k_3)`
`=k_4=(k_2k_3)/(k_2+k_3)`
Now `k_4 and k_1` are in parallel. So, equivalent spring constant `k=k_1+k_4`
`=(k_2k_3)/(k_2+k_3)+k_1`
`=(k_2k_3+k_1k_2+k_1k_3)/(k_2+k_3)`
`:. T=2pisqrt(M/k)`
`=2pi sqrt((m(k_2+k_3))/(k_2k_3+k_1k_2+k_1k_3))`
`b`. Frequency `=1/T`
`=1/(2pi)sqrt((k_2+k_3+k_1k_2+k_1k_3)/(M(k_2+k_3)))`
`c`. Amplitude =x
`=F/k=(F(k_2+k_3))/(k_1k_2+k_2k-3+k_1k_3)`