Which of the following do not have the same dimensions as the other three ? Give that l=length , m=mass , k=force constant, I= moment of inertia, B=magnetic

Options

A. `sqrt(l//g)`
B. `sqrt(I//P_mB)`
C. `sqrt(k//m)`
D. `sqrt(R//g)`

Answer

To solve the problem of determining which of the given quantities does not have the same dimensions as the others, we will analyze the dimensions of each quantity step by step.

### Step 1: Identify the quantities and their definitions
- **L**: Length
- **m**: Mass
- **k**: Force constant (spring constant)
- **I**: Moment of inertia
- **B**: Magnetic field strength

### Step 2: Write down the dimensions of each quantity
1. **Length (L)**: The dimension of length is given by:
\[ [L] = L \]

2. **Mass (m)**: The dimension of mass is given by:
\[ [m] = M \]

3. **Force constant (k)**: The force constant (spring constant) is defined as:
\[ k = \frac{F}{x} \]
where \(F\) is force and \(x\) is displacement (length). The dimension of force is:
\[ [F] = MLT^{-2} \]
Thus, the dimension of \(k\) becomes:
\[ [k] = \frac{[F]}{[x]} = \frac{MLT^{-2}}{L} = MT^{-2} \]

4. **Moment of inertia (I)**: The moment of inertia is defined as:
\[ I = m \cdot r^2 \]
where \(r\) is the distance (length). Therefore, the dimension of moment of inertia is:
\[ [I] = M \cdot L^2 \]

5. **Magnetic field strength (B)**: The magnetic field strength has the dimension:
\[ [B] = M T^{-2} A^{-1} \]
where \(A\) is the dimension of electric current.

### Step 3: Compare the dimensions
Now we will summarize the dimensions:
- **Length (L)**: \(L\)
- **Mass (m)**: \(M\)
- **Force constant (k)**: \(MT^{-2}\)
- **Moment of inertia (I)**: \(ML^2\)
- **Magnetic field strength (B)**: \(MT^{-2} A^{-1}\)

### Step 4: Identify the odd one out
Now we can compare the dimensions:
- **L** has dimension \(L\)
- **m** has dimension \(M\)
- **k** has dimension \(MT^{-2}\)
- **I** has dimension \(ML^2\)
- **B** has dimension \(MT^{-2} A^{-1}\)

From the above analysis, we can see that:
- \(k\) and \(B\) both have dimensions involving mass and time.
- \(I\) has dimensions involving mass and length squared.
- \(L\) and \(m\) are fundamentally different as one is length and the other is mass.

### Conclusion
The quantity that does not have the same dimensions as the others is **L (length)**.

### Final Answer
**L (length)** does not have the same dimensions as the other three quantities.

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