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Find the period of small vertical oscillations of a body with mass `m` in the system illustrated in figure. The stiffness values of the springs are `x_(1)` and `x_(2)`, their masses are negligible. |
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Answer» During the vertical oscillation let us locate the bock at a vertical down distance `x` from its equilibrium position . At his moment if `x_(1)` and `x_(2)` are the additional or further elongation of the upper `&` lower springs relative to the equilibrium position, then the net unbalanced force on the block will be `k_(2) x_(2)` directed in upward direction. Hence `-k_(2)x_(2)=m ddot(x) ............(1)` we also have `x=x_(1)+x_(2) ......(2)` As the springs are massless and initially the net force on the spring is also zero so for the spring `k_(1)x_(1)=k_(2)x_(2).......(3)` Solving the Eqns `(1), (2)` and `(3)` simultaneoulsy, we get `-(k_(1)k_(2))/(k_(1)+k_(2))x=m ddot(x)` Thus`ddot(x)=-((k_(1)k_(2)//k_(1)+k_(2)))/(m)x` Hence the sought time period `T=2pisqrt(m(k_(1)+k_(2))/(k_(1)k_(2)))` |
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