Out of the following the correct order of dimensions of mass increases is
(A) Velocity `" "` (B) Power
(C ) Graviational Constant

Options

A. A, B , C
B. C, A, B
C. A, C, B
D. B, C, A

Answer

To determine the correct order of dimensions of mass increases for the given quantities (Velocity, Power, and Gravitational Constant), we will analyze the dimensions of each quantity step by step.

### Step 1: Determine the dimension of Velocity
- Velocity is defined as displacement per unit time.
- The dimension of displacement (length) is denoted as \( L \) and time is denoted as \( T \).
- Therefore, the dimension of velocity is:
\[ [\text{Velocity}] = \frac{L}{T} = L T^{-1} \]

### Step 2: Determine the dimension of Power
- Power is defined as work done per unit time.
- The dimension of work is given by force times distance. The dimension of force is \( M L T^{-2} \) (from Newton's second law), and distance has the dimension \( L \).
- Thus, the dimension of work is:
\[ [\text{Work}] = [\text{Force}] \times [\text{Distance}] = (M L T^{-2}) \times L = M L^2 T^{-2} \]
- Now, since power is work done per unit time, we have:
\[ [\text{Power}] = \frac{[\text{Work}]}{[\text{Time}]} = \frac{M L^2 T^{-2}}{T} = M L^2 T^{-3} \]

### Step 3: Determine the dimension of Gravitational Constant
- The gravitational constant \( G \) can be derived from Newton's law of gravitation, which states that the gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by:
\[ F = G \frac{m_1 m_2}{r^2} \]
- Rearranging this gives:
\[ G = \frac{F r^2}{m_1 m_2} \]
- The dimension of force \( F \) is \( M L T^{-2} \), and the dimension of \( r^2 \) is \( L^2 \). The masses \( m_1 \) and \( m_2 \) both have the dimension \( M \).
- Therefore, the dimension of \( G \) is:
\[ [G] = \frac{M L T^{-2} \cdot L^2}{M^2} = M^{-1} L^3 T^{-2} \]

### Step 4: Compare the dimensions
Now we have the dimensions for each quantity:
- Velocity: \( [V] = L T^{-1} \) (mass dimension = \( M^0 \))
- Power: \( [P] = M L^2 T^{-3} \) (mass dimension = \( M^1 \))
- Gravitational Constant: \( [G] = M^{-1} L^3 T^{-2} \) (mass dimension = \( M^{-1} \))

### Step 5: Order the dimensions of mass
Now we can order them based on the dimension of mass:
1. Gravitational Constant \( G \) has the lowest mass dimension (\( M^{-1} \)).
2. Velocity \( V \) has a mass dimension of \( M^0 \).
3. Power \( P \) has the highest mass dimension (\( M^1 \)).

Thus, the correct order of the dimensions of mass increases is:
\[ G < V < P \]

### Final Answer
The correct order of dimensions of mass increases is:
**(C) Gravitational Constant < (A) Velocity < (B) Power**

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