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Discussion Response Post to: Find the maximum elongation of the spring

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anchit saini (4396)

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$\mbox{Just an extension }->\\ \\ \mbox{Irodov 1.152 - Two bars connected by a weightless spring }\\ \\ \mbox{of stiffness k and length (in the non-deformable state) }l_0 \\ \\ \mbox{ rest on a horizontal plane . A constant force F starts acting } \\ \\ \mbox{on one of the bars as shown in fig. Find max and min distances between } \\ \\ \mbox{the bars during the subsequent motion of the system} \\ \\ \\ \\ \mbox{Here we go :D } \\ \\ \mbox{Here relative velocity approach seems quite comfortable } \\ \\ \mbox{At any time the distance between the blocks is} \\ \\ l_0 + x \mbox{ where x is the elongation in spring} \\ \\ \mbox{Equations of motion} -> \\ \\ kx = m_1 a_1 \\ \\ F - kx = m_2 a_2 \\ \\ a_r= \mbox{rel. accn. of 2 with respect to 1}\\ \\ =\frac{F-kx}{m_2} -\frac{kx}{m_1} \\ \\ a_r=v\frac{dv}{dx}\\ \\ -> \\ \\ \int{vdv}=\int{[\frac{F-kx}{m_2} -\frac{kx}{m_1}] dx}\\ \\ \mbox{Since initially v relative is 0 and finally also its 0 (cos both the blocks}\\ \\ \mbox{ with same velocity during max separation), the limits of L.H.S are from 0 to 0}\\ \\ \mbox{while limits of R.H.S are from }l_0 \ to \ l_{max} \\ \\ \\ \\ \mbox{On solving , we get}-> \\ \\ l_{max}=l_0 + \frac{2m_1F}{k(m_1+m_2)}$

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