In a book, the answer for a particular question is expressed as `b=(ma)/(k)[sqrt(1+(2kl)/(ma))]` here m represents mass, a represent acceleration, l represent length. The unit of b should be :-

Options

A. `m//s`
B. `m//s^(2)`
C. meter
D. /sec

Answer

To find the unit of \( b \) given the expression

\[ b = \frac{ma}{k} \sqrt{1 + \frac{2kl}{ma}} \]

we will analyze the dimensions of each component step by step.

### Step 1: Identify the dimensions of each variable
- **Mass (m)** has the dimension \([M]\).
- **Acceleration (a)** has the dimension \([L T^{-2}]\).
- **Length (l)** has the dimension \([L]\).
- **k** is an unknown constant, and we will determine its dimensions later.

### Step 2: Analyze the term \( \frac{ma}{k} \)
We need to find the dimensions of \( \frac{ma}{k} \).

1. The dimension of \( ma \) is:
\[ [m] \cdot [a] = [M] \cdot [L T^{-2}] = [M L T^{-2}] \]

2. Now, let's denote the dimension of \( k \) as \([K]\). Therefore, the dimension of \( \frac{ma}{k} \) is:
\[ \frac{[M L T^{-2}]}{[K]} = \frac{[M L T^{-2}]}{[K]} \]

### Step 3: Analyze the term \( \frac{2kl}{ma} \)
Next, we need to analyze \( \frac{2kl}{ma} \).

1. The dimension of \( kl \) is:
\[ [K] \cdot [L] = [K L] \]

2. The dimension of \( ma \) is already calculated as \([M L T^{-2}]\).

3. Therefore, the dimension of \( \frac{2kl}{ma} \) is:
\[ \frac{[K L]}{[M L T^{-2}]} = \frac{[K]}{[M T^{-2}]} \]

### Step 4: Set the dimension of \( 1 + \frac{2kl}{ma} \)
For the expression \( 1 + \frac{2kl}{ma} \) to be dimensionless, the dimensions of \( \frac{2kl}{ma} \) must also be dimensionless. This implies:
\[ [K] = [M T^{-2}] \]

### Step 5: Substitute back to find the dimension of \( \frac{ma}{k} \)
Now we can substitute back to find the dimension of \( \frac{ma}{k} \):
\[ \frac{ma}{k} = \frac{[M L T^{-2}]}{[M T^{-2}]} = [L] \]

### Step 6: Analyze the square root term
The term \( \sqrt{1 + \frac{2kl}{ma}} \) is dimensionless, so it does not contribute any dimensions.

### Step 7: Combine the results
Thus, the dimension of \( b \) is:
\[ b = \frac{ma}{k} \sqrt{1 + \frac{2kl}{ma}} = [L] \cdot [1] = [L] \]

### Conclusion
The unit of \( b \) is the unit of length, which is typically expressed in meters (m).

### Final Answer
The unit of \( b \) should be: **meters (m)**.