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JEE Main & Advanced Sample Paper JEE Main - Mock Test - 9

  • question_answer
    A new system of units is proposed in which unit of mass is \[\alpha \] kg, unit of length is \[\beta \,m\]and unit of time is \[\gamma \,s\]. What will be value of 5 J in this new system?

    A) \[5\alpha {{\beta }^{2}}{{\gamma }^{-2}}\]   

    B) \[5{{\alpha }^{-1}}{{\beta }^{-2}}{{\gamma }^{-2}}\]

    C) \[5{{\alpha }^{-2}}{{\beta }^{-1}}{{\gamma }^{-2}}\]          

    D) \[5{{\alpha }^{-1}}{{\beta }^{2}}{{\gamma }^{2}}\]

    Correct Answer: B

    Solution :

    Joule is a unit of ernergy.
    SI New system
    \[{{n}_{1}}=5\] \[{{n}_{2}}=?\]
    \[{{M}_{1}}=1\,kg\] \[{{M}^{2}}=\alpha \,kg\]
    \[{{L}_{1}}=1\,m\] \[{{L}^{2}}=\beta \,m\]
    \[{{T}_{1}}=1s\] \[{{T}^{2}}=\gamma s\]
    Dimensional formula of energy is comparing with, \[[{{M}^{a}}{{L}^{b}}{{T}^{c}}],\]we get \[a=1,\,b=2,\,c=-2\] As \[{{n}_{2}}={{n}_{1}}{{\left( \frac{{{M}^{1}}}{{{M}^{2}}} \right)}^{a}}{{\left( \frac{{{L}^{1}}}{{{L}^{2}}} \right)}^{b}}{{\left( \frac{{{T}_{1}}}{{{T}_{2}}} \right)}^{c}}\]\[=5{{\left( \frac{1\,kg}{\alpha \,kg} \right)}^{1}}{{\left( \frac{1m}{\beta m} \right)}^{2}}{{\left( \frac{1s}{\gamma s} \right)}^{-2}}=\frac{5{{\gamma }^{2}}}{\alpha {{\beta }^{2}}}=\frac{5{{\gamma }^{2}}}{\alpha {{\beta }^{2}}}=5{{\alpha }^{-1}}{{\beta }^{-2}}{{\gamma }^{2}}\]


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