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Blazing goIITian

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5 Sep 2008 11:57:33 IST

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Find the maximum elongation of the spring

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Find the maximum elongation of the spring

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VARUN RAJ

Blazing goIITian

Joined: 16 Mar 2008

Posts: 1825

5 Sep 2008 12:41:37 IST

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THE MAXIMUM ELINGATION OF THE SPRING IS WHEN THE BLOCKS HV THE SAME VELCOITY
m1V=(m1+m2)V2
V2=M1V/M1+M2
SINCE NO EXT FORCE ACTS ON THE SYSTEM THE ENERGY WILL BE CONSERVED
SO LOSS IN K.E.=GAIN IN SPRING ENRGY
SOLVE AND GET THE ANSWER
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anchit saini

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5 Sep 2008 12:56:50 IST

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$\mbox{Max compression would be when both the blocks move with common velocity =} \\ \\ v_{com} =\frac{m_1v}{m_1 + m_2} \\ \\ Initial \ energy \ = \frac{m_1v^2}{2} \\ \\ = final \ energy =\frac{kx^2}{2} + \frac{m_1v_{com}^2}{2} +\frac{m_2v_{com}^2}{2} \\ \\ On \ solving -> \\ \\ x=\sqrt{\frac{m_1m_2v^2}{k(m_1+m_2)}}$

anchit saini

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5 Sep 2008 12:57:25 IST

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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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