figure shown a partical mass `m = 100g` attaches with four identical spring , each of length `l = 10 cm` . Initial tension in each spring is `F_(0) = 25 N`. Neglecting gravity , Calculate the period of small oscillation of the article along a line perpendicular to the plane of the figure .

Explanation

Let tha prtical be displaced singhtly through x along a line normal to the plane of figure . Then each spring is further elongated . Since , springs are indentical , therefore increase in tension of each spring will be the same. Let this increase be `df_(0)`

First consider force exerted by spring `AP and CP` only as shown in figure
Restoring force produced by these two spring `= (F_(0) + dF_(0)) 22 sin theta`.
Since x is very small, hterefore , `sin theta = x ll`.
Neglecting produced of veru small quantities, restoring force produced by these two springs `= 2 F_(0) x ll`
Similarly, resorting force produced by two remaining springs `BP and DP` will also be equal to `(2 F_(0) x ll)`
Resultant resoring force.
`F = 2 xx ((2 F_(0) x)/(l)) = (4F_(0))/(l) x`
Restoring acceleration is directly proportinal to displacement `x`, therefore ,the partical executes SHM
Its period `T = 2 pi sqrt(("displacement")/("acceleration"))`
`= 2 pi sqrt((ml)/(4 F_(0))) = pi sqrt((ml)/(4 F_(0)) = 0.02 pi s`