Which one of the following have same dimensions :

Options

A. Potential energy and force
B. Torque and force
C. Torque and potential energy
D. Planck's constant and momentum.

Answer

To determine which of the given options have the same dimensions, we need to analyze the dimensions of each physical quantity mentioned. Let's break down the options step by step.

### Step 1: Analyze Potential Energy
Potential energy (PE) is given by the formula:
\[ PE = mgh \]
Where:
- \( m \) = mass (dimension: \([M]\))
- \( g \) = acceleration due to gravity (dimension: \([LT^{-2}]\))
- \( h \) = height (dimension: \([L]\))

Thus, the dimensions of potential energy are:
\[ [PE] = [M][LT^{-2}][L] = [ML^2T^{-2}] \]

### Step 2: Analyze Force
Force (F) is given by Newton's second law:
\[ F = ma \]
Where:
- \( m \) = mass (dimension: \([M]\))
- \( a \) = acceleration (dimension: \([LT^{-2}]\))

Thus, the dimensions of force are:
\[ [F] = [M][LT^{-2}] = [MLT^{-2}] \]

### Step 3: Compare Potential Energy and Force
From the above calculations:
- Dimensions of Potential Energy: \([ML^2T^{-2}]\)
- Dimensions of Force: \([MLT^{-2}]\)

Since the dimensions are not the same, potential energy and force do not have the same dimensions.

### Step 4: Analyze Torque
Torque (\(\tau\)) is given by:
\[ \tau = F \cdot d \]
Where:
- \( F \) = force (dimension: \([MLT^{-2}]\))
- \( d \) = distance (dimension: \([L]\))

Thus, the dimensions of torque are:
\[ [\tau] = [MLT^{-2}][L] = [ML^2T^{-2}] \]

### Step 5: Compare Torque and Potential Energy
From the calculations:
- Dimensions of Torque: \([ML^2T^{-2}]\)
- Dimensions of Potential Energy: \([ML^2T^{-2}]\)

Since the dimensions are the same, torque and potential energy have the same dimensions.

### Step 6: Analyze Planck's Constant
Planck's constant (\(h\)) has dimensions:
\[ [h] = [ML^2T^{-1}] \]

### Step 7: Analyze Momentum
Momentum (\(p\)) is given by:
\[ p = mv \]
Where:
- \( m \) = mass (dimension: \([M]\))
- \( v \) = velocity (dimension: \([LT^{-1}]\))

Thus, the dimensions of momentum are:
\[ [p] = [M][LT^{-1}] = [MLT^{-1}] \]

### Step 8: Compare Planck's Constant and Momentum
From the calculations:
- Dimensions of Planck's Constant: \([ML^2T^{-1}]\)
- Dimensions of Momentum: \([MLT^{-1}]\)

Since the dimensions are not the same, Planck's constant and momentum do not have the same dimensions.

### Conclusion
The only pair of quantities that have the same dimensions are torque and potential energy.

**Final Answer: C (Torque and Potential Energy)**