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Question: A calorie is a unit of heat or energy and it equals about \[4.2J\] where\(1J = 1kg{m^2}{s^{ - 2}}\)....

A calorie is a unit of heat or energy and it equals about 4.2J4.2J where1J=1kgm2s−21J = 1kg{m^2}{s^{ - 2}}. Suppose we employ a system of units in which the units of mass equal α kg\alpha {\text{ }}kg, the units of length equals β m\beta {\text{ }}m, the unit of time isγ s\gamma {\text{ s}}. Show that a calorie has a magnitude4.2α−1β−2γ24.2{\alpha ^{ - 1}}{\beta ^{ - 2}}{\gamma ^2} in terms of the new units.

Explanation

Solution

Here we will be using the concept of one of the applications of dimensional analysis, that is converting one system of unit into another system of unit. First, the powers of the dimensions have to be assumed as some unknown factors and ten find their values by using the actual dimensions. Next, put the given values and find the required magnitude.

Complete step by step answer:
The dimension of any physical quantity can be written as the product of three principal dimensions having some powers.
And , the standard formula of unit conversion states that,
given unitnew unit=(M1M2)x(L1L2)y(T1T2)z\dfrac{{{\text{given unit}}}}{{{\text{new unit}}}} = {\left( {\dfrac{{{M_1}}}{{{M_2}}}} \right)^x}{\left( {\dfrac{{{L_1}}}{{{L_2}}}} \right)^y}{\left( {\dfrac{{{T_1}}}{{{T_2}}}} \right)^z}
Here the given unit is calorie.
1 calorie = 4.2J1{\text{ calorie = 4}}{\text{.2J}} and 1 J = 1 kgm2s−21{\text{ J = 1 kg}}{{\text{m}}^2}{s^{ - 2}}
In S.I system, M1=1kg,L1=1m,T1=1s{M_1} = 1kg,{L_1} = 1m,{T_1} = 1s
And in the new system given in the problem, M2=αkg,L2=βm,T2=γs{M_2} = \alpha kg,{L_2} = \beta m,{T_2} = \gamma s
Hence, Calorienew unit=4.2(1α)x(1β)y(1γ)z\dfrac{{Calorie}}{{{\text{new unit}}}} = 4.2{\left( {\dfrac{1}{\alpha }} \right)^x}{\left( {\dfrac{1}{\beta }} \right)^y}{\left( {\dfrac{1}{\gamma }} \right)^z}
Since the calorie is unit of heat energy and the dimension of the heat is [M1L2T−2][{M^1}{L^2}{T^{ - 2}}]
∴x=1,y=2,z=−2\therefore x = 1,y = 2,z = - 2
∴Calorienew unit=4.2(1α)1(1β)2(1γ)−2\therefore \dfrac{{Calorie}}{{{\text{new unit}}}} = 4.2{\left( {\dfrac{1}{\alpha }} \right)^1}{\left( {\dfrac{1}{\beta }} \right)^2}{\left( {\dfrac{1}{\gamma }} \right)^{ - 2}}
⇒Calorie=α−1β−2γ2new unit\Rightarrow Calorie = {\alpha ^{ - 1}}{\beta ^{ - 2}}{\gamma ^2}{\text{new unit}}

Additional information:
Dimensional formulae can be used to convert one system of the unit to another system of the unit.
Dimensional formulae can be used to check the correctness of a given equation.
Dimensional formulae can be used to derive the relationship among different physical quantities.

Note: Heat is a form of energy called thermal energy. When we bring a hot body in thermal contact with a cold body, the temperature of the hot body decreases, and the temperature of the cold body increases. The energy of the hot body decreases and that of the cold body increases. This transfer of energy takes place until the two bodies attain a common temperature. So, heat flows from a body at a higher temperature to a body at a lower temperature. Si unit of heat energy is the joule.
A calorie is the most commonly used unit of heat. A calorie is generally defined as the amount of heat required by one gram of water to raise its temperature by one degree Celsius.