A new system of units is proposed in which unit of mass is `alpha kg,` unit of length `beta` m and unit of time `gamma s.` How much will 5 J measure in this new system ?

Options

A. `5alphabeta^(2)gamma^(-2)`
B. `5alpha^(-1)beta^(-2)gamma^(2)`
C. `5alpha^(-2)beta^(-1)gamma^(-2)`
D. `5alpha^(-1)beta^(2)gamma^(2)`

Explanation

To solve the problem of converting 5 Joules into the new system of units defined by `alpha kg` for mass, `beta m` for length, and `gamma s` for time, we will follow these steps:

### Step 1: Understand the SI Units
First, we need to recall the SI units for Joules (J).
1 Joule is defined as:
\[ 1 \text{ J} = 1 \text{ kg} \cdot \text{m}^2/\text{s}^2 \]
This means that in terms of dimensions, energy (Joules) can be expressed as:
\[ \text{Energy} = [M^1 L^2 T^{-2}] \]
where \(M\) is mass, \(L\) is length, and \(T\) is time.

### Step 2: Define the New Units
In the new system:
- The unit of mass is `alpha kg` (let's denote it as \(M_2 = \alpha\))
- The unit of length is `beta m` (let's denote it as \(L_2 = \beta\))
- The unit of time is `gamma s` (let's denote it as \(T_2 = \gamma\))

### Step 3: Write the Dimensional Formula for Energy
The dimensional formula for energy in terms of the new units will be:
\[ N_2 = N_1 \left( \frac{L_1}{L_2} \right)^a \left( \frac{M_1}{M_2} \right)^b \left( \frac{T_1}{T_2} \right)^c \]
where \(N_1\) is the energy in the original system (5 J), and \(a\), \(b\), and \(c\) are the powers corresponding to length, mass, and time respectively.

From the dimensional formula of energy:
- \(a = 2\) (for length)
- \(b = 1\) (for mass)
- \(c = -2\) (for time)

### Step 4: Substitute the Values
Now, substituting the values into the equation:
\[ N_2 = 5 \left( \frac{1}{\beta} \right)^2 \left( \frac{1}{\alpha} \right)^1 \left( \frac{1}{\gamma} \right)^{-2} \]

This simplifies to:
\[ N_2 = 5 \cdot \frac{1}{\beta^2} \cdot \frac{1}{\alpha} \cdot \gamma^2 \]

### Step 5: Rearranging the Equation
Rearranging gives us:
\[ N_2 = 5 \cdot \frac{\gamma^2}{\alpha \beta^2} \]

### Final Expression
Thus, the expression for 5 Joules in the new system of units is:
\[ N_2 = 5 \alpha^{-1} \beta^{-2} \gamma^2 \]

### Conclusion
The final answer is:
\[ N_2 = 5 \alpha^{-1} \beta^{-2} \gamma^2 \]