The fundamental physical quantities that have same dimensions in the dimensional formula of torque and Angular momentum are

Options

A. mass, time
B. time, length
C. mass, length
D. time , mole

Answer

To determine the fundamental physical quantities that have the same dimensions in the dimensional formula of torque and angular momentum, we can follow these steps:

### Step 1: Understand the definition of Torque
Torque (τ) is defined as the product of force (F) and the distance (r) from the pivot point. The formula for torque is:
\[ \tau = F \cdot r \]

### Step 2: Find the dimensional formula of Force
Force is defined by Newton's second law as mass (m) times acceleration (a). The dimensional formula for acceleration is:
\[ a = \frac{L}{T^2} \]
Thus, the dimensional formula for force is:
\[ F = m \cdot a = m \cdot \frac{L}{T^2} = [M^1 L^1 T^{-2}] \]

### Step 3: Find the dimensional formula of Torque
Now substituting the dimensional formula of force into the torque formula:
\[ \tau = F \cdot r = [M^1 L^1 T^{-2}] \cdot [L^1] = [M^1 L^2 T^{-2}] \]
So, the dimensional formula for torque is:
\[ [\tau] = [M^1 L^2 T^{-2}] \]

### Step 4: Understand the definition of Angular Momentum
Angular momentum (L) is defined as the product of mass (m) and velocity (v) multiplied by the radius (r):
\[ L = m \cdot v \cdot r \]

### Step 5: Find the dimensional formula of Velocity
The dimensional formula for velocity is:
\[ v = \frac{L}{T} = [L^1 T^{-1}] \]

### Step 6: Find the dimensional formula of Angular Momentum
Now substituting the dimensional formula of mass and velocity into the angular momentum formula:
\[ L = m \cdot v \cdot r = [M^1] \cdot [L^1 T^{-1}] \cdot [L^1] = [M^1 L^2 T^{-1}] \]
So, the dimensional formula for angular momentum is:
\[ [Angular Momentum] = [M^1 L^2 T^{-1}] \]

### Step 7: Compare the dimensional formulas
Now we can compare the dimensional formulas of torque and angular momentum:
- Torque: \([M^1 L^2 T^{-2}]\)
- Angular Momentum: \([M^1 L^2 T^{-1}]\)

### Step 8: Identify common dimensions
From the comparison, we can see that both torque and angular momentum share the same dimensions for mass and length:
- Mass: \(M^1\)
- Length: \(L^2\)

### Conclusion
The fundamental physical quantities that have the same dimensions in the dimensional formula of torque and angular momentum are **mass and length**.