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Question: The unit of Mass is \[\alpha \text{ kg}\]. The unit of length is \[\beta \text{ metre}\] and the uni...

The unit of Mass is α kg\alpha \text{ kg}. The unit of length is β metre\beta \text{ metre} and the unit of time is γ second\gamma \text{ second}. The magnitude of calorie in the new system is: [1 calorie = 4.2 Joules]\left[ \text{1 calorie = 4}\text{.2 Joules} \right]
(A) 4.2α1β2γ−2 new units4.2{{\alpha }^{1}}{{\beta }^{2}}{{\gamma }^{-2}}\text{ new units}
(B) 4.2α−1β−2γ2 new units4.2{{\alpha }^{-1}}{{\beta }^{-2}}{{\gamma }^{2}}\text{ new units}
(C) α−1β−2γ2 new units{{\alpha }^{-1}}{{\beta }^{-2}}{{\gamma }^{2}}\text{ new units}
(D) 14.2α−1β−2γ2 new units\dfrac{1}{4.2}{{\alpha }^{-1}}{{\beta }^{-2}}{{\gamma }^{2}}\text{ new units}

Explanation

Solution

We can approach this problem like a problem of dimensional analysis. Both Joule and Calorie are units of energy or the work done. Since energy is a derived physical quantity, we can express it in terms of basic physical units given to us, that is, mass, length and time and we can find the correct answer.

Formula Used:
Work done = Force × displacement\text{Work done = Force }\times \text{ displacement} ,

Complete step by step solution:
As discussed in the hint, Joule is a unit of work done or energy spent.
We know that, Work done = Force × displacement\text{Work done = Force }\times \text{ displacement}
Force can be further broken down as Force = mass × acceleration\text{Force = mass }\times \text{ acceleration}
Combining the two relations mentioned above, we get Work done =mass × acceleration× displacement\text{Work done =mass }\times \text{ acceleration}\times \text{ displacement}
We have reduced the work done to only one derived unit, which is acceleration
Acceleration can be represented as velocity per unit time and velocity is the displacement per unit time, this means that acceleration can be written as acceleration = displacement(time)2\text{acceleration = }\dfrac{\text{displacement}}{{{(\text{time)}}^{\text{2}}}}
Finally we can write work done as Work done =mass ×displacement(time)2× displacement\text{Work done =mass }\times \dfrac{\text{displacement}}{{{(\text{time)}}^{\text{2}}}}\times \text{ displacement}
Displacement has the unit of length and is, as such, a basic physical quantity
We have been given the units of length, mass and time as β metre\beta \text{ metre} , α kg\alpha \text{ kg} and γ second\gamma \text{ second} respectively
Substituting the values in the expression for work done, we get

& \text{work done = }\alpha \text{ kg}\times {{\left( \dfrac{\beta \text{ metre}}{\gamma \text{ second}} \right)}^{2}} \\\ & \Rightarrow \text{work done = }{{\alpha }^{1}}{{\beta }^{2}}{{\gamma }^{-2}}\text{ Joules} \\\ \end{aligned}$$ We have been asked to express the work done in calories We have been told that $$\text{1 calorie = 4}\text{.2 Joules}$$ Hence we can say that work done in calories will be $$4.2{{\alpha }^{1}}{{\beta }^{2}}{{\gamma }^{-2}}\text{ new units}$$. **Therefore option (A) is the correct answer.** **Note:** There are typically two types of unit conversions. Some conversions from one system of units to another need to be exact, without increasing or decreasing the precision of the first measurement. This is called soft conversion. It does not involve changing the physical configuration of the item being measured. By contrast, a hard conversion or an adaptive conversion may not be exactly equivalent. It changes the measurement to convenient and workable numbers and units in the new system. The given question makes use of soft conversion. Also, if you remember the dimensional formula of energy, which is $$[M{{L}^{2}}{{T}^{-2}}]$$ , you can easily find the correct answer without doing any calculations or any quantity conversion.