Time (T), velocity (C) and angular momentum (h) are chosen as fundamental quantities instead of mass, length and time. In terms of these, the dimensions of mass would be

Options

A. `[M] = [T^(-1)C^(-2)h]`
B. `[M] = [TC^(-2) h]`
C. `[M] = [T^(-1)C^(-2)h^(-1)]`
D. `[M] = [T^(-1) C^(2)h]`

Answer

To find the dimensions of mass in terms of the fundamental quantities time (T), velocity (C), and angular momentum (H), we will follow these steps:

### Step 1: Understand the relationship between mass, time, velocity, and angular momentum.
We know that mass can be expressed in terms of the fundamental quantities. We can write:
\[ m \propto T^a C^b H^c \]
where \( a \), \( b \), and \( c \) are the powers we need to determine.

### Step 2: Write the dimensions of the quantities involved.
- The dimension of time \( T \) is \( [T] \).
- The dimension of velocity \( C \) is \( [L T^{-1}] \) (length per unit time).
- The dimension of angular momentum \( H \) is given by the formula \( H = m \cdot v \cdot r \), which can be expressed as:
\[ H = [M][L T^{-1}][L] = [M][L^2 T^{-1}] \]

### Step 3: Substitute the dimensions into the proportionality equation.
Now substituting the dimensions into our equation:
\[ [M] \propto [T]^a \cdot ([L T^{-1}])^b \cdot ([M][L^2 T^{-1}])^c \]

This expands to:
\[ [M] \propto [T]^a \cdot [L^b T^{-b}] \cdot [M^c L^{2c} T^{-c}] \]

### Step 4: Combine the dimensions.
Combining the dimensions gives us:
\[ [M] \propto [M^c][L^{b + 2c}][T^{a - b - c}] \]

### Step 5: Equate the dimensions.
Since we want to find the dimensions of mass, we can equate the dimensions on both sides:
1. Coefficient of \( [M] \): \( 1 = c \)
2. Coefficient of \( [L] \): \( 0 = b + 2c \)
3. Coefficient of \( [T] \): \( 0 = a - b - c \)

### Step 6: Solve the equations.
From \( 1 = c \), we get:
\[ c = 1 \]

Substituting \( c = 1 \) into \( 0 = b + 2c \):
\[ 0 = b + 2(1) \]
\[ b = -2 \]

Now substituting \( b = -2 \) and \( c = 1 \) into \( 0 = a - b - c \):
\[ 0 = a - (-2) - 1 \]
\[ 0 = a + 2 - 1 \]
\[ a = -1 \]

### Step 7: Write the final expression for mass.
Now we have:
- \( a = -1 \)
- \( b = -2 \)
- \( c = 1 \)

Thus, the dimensions of mass in terms of \( T \), \( C \), and \( H \) are:
\[ [M] = T^{-1} C^{-2} H^{1} \]

### Final Answer:
The dimensions of mass in terms of \( T \), \( C \), and \( H \) are:
\[ [M] = T^{-1} C^{-2} H^{1} \]

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