Which of the following have same dimensions

Options

A. angular momentum and linear momentum
B. work and power
C. work and torque
D. torque and pressure.

Answer

To determine which of the given quantities have the same dimensions, we will analyze the dimensions of each quantity step by step.

### Step 1: Analyze Angular Momentum
Angular momentum (L) is given by the formula:
\[ L = I \cdot \omega \]
Where:
- \( I \) is the moment of inertia, which has dimensions of:
\[ [I] = [M][L^2] \]
- \( \omega \) is the angular velocity, which has dimensions of:
\[ [\omega] = [T^{-1}] \]

Thus, the dimensions of angular momentum are:
\[ [L] = [M][L^2][T^{-1}] = [M][L^2][T^{-1}] \]

### Step 2: Analyze Linear Momentum
Linear momentum (p) is given by the formula:
\[ p = m \cdot v \]
Where:
- \( m \) is mass with dimensions:
\[ [m] = [M] \]
- \( v \) is velocity, which has dimensions of:
\[ [v] = [L][T^{-1}] \]

Thus, the dimensions of linear momentum are:
\[ [p] = [M][L][T^{-1}] \]

### Step 3: Compare Angular Momentum and Linear Momentum
From the above calculations:
- Dimensions of angular momentum: \( [M][L^2][T^{-1}] \)
- Dimensions of linear momentum: \( [M][L][T^{-1}] \)

Since \( [L^2] \) and \( [L] \) are not the same, angular momentum and linear momentum do not have the same dimensions.

### Step 4: Analyze Work and Power
Work (W) is given by:
\[ W = F \cdot d \]
Where:
- \( F \) is force, which has dimensions of:
\[ [F] = [M][L][T^{-2}] \]
- \( d \) is displacement with dimensions:
\[ [d] = [L] \]

Thus, the dimensions of work are:
\[ [W] = [M][L][T^{-2}][L] = [M][L^2][T^{-2}] \]

Power (P) is defined as:
\[ P = \frac{W}{t} \]
Where \( t \) has dimensions of:
\[ [t] = [T] \]

Thus, the dimensions of power are:
\[ [P] = \frac{[M][L^2][T^{-2}]}{[T]} = [M][L^2][T^{-3}] \]

### Step 5: Compare Work and Power
From the above calculations:
- Dimensions of work: \( [M][L^2][T^{-2}] \)
- Dimensions of power: \( [M][L^2][T^{-3}] \)

Since the time dimensions are different, work and power do not have the same dimensions.

### Step 6: Analyze Torque
Torque (τ) is defined as:
\[ \tau = r \times F \]
Where:
- \( r \) is the distance with dimensions:
\[ [r] = [L] \]
- \( F \) is force, which we already established has dimensions of:
\[ [F] = [M][L][T^{-2}] \]

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

### Step 7: Compare Work and Torque
From the above calculations:
- Dimensions of work: \( [M][L^2][T^{-2}] \)
- Dimensions of torque: \( [M][L^2][T^{-2}] \)

Since both have the same dimensions, work and torque have the same dimensions.

### Conclusion
The quantities that have the same dimensions are **Work and Torque**.

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