If mass is measured in unit of `alpha` kg, length in `beta` m and time in `gamma ` s, then calorie would be

Options

A. `4 2 alpha beta^(2)gamma^(-2)`
B. `[MT^(-2)]`
C. `42alpha^(-1)beta^(-2)gamma^(-2)`
D. `42alpha^(-2)beta^(-2)gamma^(-2)`

Answer

To determine the unit of calorie when mass is measured in `alpha` kg, length in `beta` m, and time in `gamma` s, we need to start by understanding the relationship between calories and joules, as well as how to express joules in terms of the given units.

### Step-by-Step Solution:

1. **Understand the definition of Joule**:
The SI unit of energy is the Joule (J). One Joule is defined as the work done when a force of one newton displaces an object by one meter in the direction of the force. Mathematically, it can be expressed as:
\[ 1 \text{ Joule} = 1 \text{ Newton} \times 1 \text{ meter} \]

2. **Express Newton in terms of base units**:
A newton (N) is defined as the force required to accelerate a mass of one kilogram at a rate of one meter per second squared. Therefore:
\[ 1 \text{ Newton} = 1 \text{ kg} \cdot \text{m/s}^2 \]

3. **Substituting Newton into the Joule equation**:
Now substituting the expression for Newton into the Joule definition:
\[ 1 \text{ Joule} = 1 \text{ kg} \cdot \text{m/s}^2 \cdot 1 \text{ m} = 1 \text{ kg} \cdot \text{m}^2/\text{s}^2 \]

4. **Convert Joules to calories**:
We know that:
\[ 1 \text{ calorie} = 4.184 \text{ Joules} \]
Therefore, we can express calories in terms of Joules:
\[ 1 \text{ calorie} = 4.184 \text{ kg} \cdot \text{m}^2/\text{s}^2 \]

5. **Substituting the given units**:
Now, substituting the given units where mass is `alpha` kg, length is `beta` m, and time is `gamma` s:
\[ 1 \text{ calorie} = 4.184 \cdot \alpha \cdot \beta^2/\gamma^2 \]

6. **Final expression**:
Thus, the unit of calorie in terms of the new units is:
\[ \text{calorie} = 4.184 \cdot \alpha \cdot \frac{\beta^2}{\gamma^2} \]

### Summary:
The calorie, when mass is measured in `alpha` kg, length in `beta` m, and time in `gamma` s, is given by:
\[ \text{calorie} = 4.184 \cdot \alpha \cdot \frac{\beta^2}{\gamma^2} \]