આકૃતિમાં દર્શાવ્યા પ્રમાણે $2 kg$ દળના બ્લોકને $P$ સ્થળેથી મુક્ત કરવામાં આવે છે. તે સમતલ પર $0.5 m$ સુધી સરક્યા બાદ સ્પ્રિંગ સાથે અથડાય છે.આ સ્પ્રિંગનો બળ અચળાંક $4000 N/m $ છે. બ્લોક અને ઢોળાવવાળા સમતલ વચ્ચેનો ઘર્ષણાંક $0.3 $ છે. સ્પ્રિંગમાં થતું સંકોચન ............... $\mathrm{mm}$ હશે.

Question

A$45$
B$32.54$
C$25.20$
D$34.67 $

Diffcult

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Solution

d
${W_c}\, + \,{W_{nc}}\, + \,\,{W_{ext}}\,\, = \,\,\,\Delta KE$

$ \Rightarrow \,\,\,mg(0.5\,\, + \,\,x)\sin {30^ \circ }\, - \,\,\mu mg\,(0.5\,\, + \,\,x)\,\cos {30^ \circ }\, - \,\,\frac{1}{2}k{x^2}\, = \,\,0$

$ \Rightarrow \,\,\,2\,\, \times \,\,10(0.5\,\, + \,\,x)\,\frac{1}{2}\,\, - \,\,0.3\,\, \times \,\,2\,\, \times \,\,10\,\,(0.5\,\, + \,\,x)\,\frac{{\sqrt 3 }}{2}\,\, - \,\,\frac{1}{2}\,\, \times \,\,4000{x^2}\, = \,\,0$

$ \Rightarrow \,\,20(0.5\,\, + \,\,x)\,\, - \,\,6(0.5\,\, + \,\,x)\,\sqrt 3 \,\, - \,\,4000\,{x^2}\, = \,\,0$

$ \Rightarrow \,\,\,10\,\, + \,\,20x\,\, - \,\,(3\,\, + \,\,6x)\,\sqrt 3 \,\, - \,\,4000\,\,{x^2}\, = \,\,0$

$ \Rightarrow \,\,\,10\,\, + \,\,20x\,\, - \,\,3\,\,\sqrt 3 \,\, - \,\,6\,\,\sqrt 3 \,x\,\, - \,\,4000{x^2}\, = \,\,0$

$ \Rightarrow \,\,\,4.8\,\, + \,\,9.6x\,\, - \,\,4000\,\,{x^2}\, = \,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,4000\,\,{x^2}\, - \,\,0.6\,\,x\,\, - \,\,4.8\,\, = \,\,0$

$x\,\, = \,\,\,\frac{{9.6\,\, \pm \,\,\,\sqrt {{{(9.6)}^2}\, + \,\,4(4.8)\,\,(4000)} }}{{2\,\, \times \,\,4000}}\,\, = \,\,0.03467\,\,m\,\, = \,\,34.67\,\,\,mm$

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