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Units and Measurement

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Question : 3 of 33
Marks: +1, -0
calorie is a unit of heat or energy and it equals about 4.2 J where 1 J = 1 kg m2^{2} s−2^{-2}. Suppose we employ a system of units in which the unit of mass equals α\alpha kg, the unit of length equals β\beta m, the unit of time is γ\gamma s. Show that a calorie has a magnitude 4.2 α−1β−2γ2\alpha^{-1} \beta^{-2} \gamma^{2} in terms of the new units.
Solution:  
We know that n1u1=n2u2n_{1} u_{1}=n_{2} u_{2}
or     n2=n1u1u2\;\; n_{2}=n_{1} \frac{u_{1}}{u_{2}}
=n1[M1aL1bT1c][M2aL2bT2c]=n_{1} \frac{[M_{1}^{a} L_{1}^{b} T_{1}^{c}]}{[M_{2}^{a} L_{2}^{b} T_{2}^{c}]}
=n1[M1M2]a[L1L2]b[T1T2]c=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}
1 cal =4.2 J=4.2 kg m2 s−2=4.2\,\mathrm{J}=4.2\,\mathrm{kg}\,\mathrm{m}^{2}\,\mathrm{s}^{-2}
∴a=1,  b=2,  c=−2\therefore a=1,\; b=2,\; c=-2
 SI system   New system
 n1=4.2n_{1}=4.2  n2=?n_{2}=?
 M1=1 kgM_{1}=1\,\mathrm{kg}  M2=α kgM_{2}=\alpha\,\mathrm{kg}
 L1=1 mL_{1}=1\,\mathrm{m}  L2=β mL_{2}=\beta\,\mathrm{m}
 T1=1 sT_{1}=1\,\mathrm{s}  L2=γ sL_{2}=\gamma\,\mathrm{s}
∴  n2=4.2[1 kgα kg]1[1 mβ m]2[1 sγ s]−2\therefore\; n_{2}=4.2 \left[\frac{1\,\mathrm{kg}}{\alpha\,\mathrm{kg}}\right]^{1} \left[\frac{1\,\mathrm{m}}{\beta\,\mathrm{m}}\right]^{2} \left[\frac{1\,\mathrm{s}}{\gamma\,\mathrm{s}}\right]^{-2}
∴  n2=4.2α−1β−2γ2\therefore\; n_{2}=4.2 \alpha^{-1} \beta^{-2} \gamma^{2}
∴  1\therefore\; 1 cal =4.2α−1β−2γ2=4.2 \alpha^{-1} \beta^{-2} \gamma^{2} in new system
Hence proved.
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