Find the dimensions of
a. angular speed `omega`
angular acceleration `alpha`
torque `tau` and
d. moment of interia `I`.
. Some of the equations involving these quantities are `omega=(theta_2-theta_1)/(t_2-t_1), alpha = (omega_2-omega_1)/(t_2-t_1), tau= F.r and I=mr^2`
The symbols have standard meanings.

Explanation

To find the dimensions of the given physical quantities, we will analyze each one step by step based on the provided equations.

### Step 1: Find the dimensions of angular speed (ω)

**Definition**: Angular speed (ω) is defined as the rate of change of angle with respect to time.

**Formula**:
\[ \omega = \frac{\theta_2 - \theta_1}{t_2 - t_1} \]

- The angle (θ) is a dimensionless quantity (it is measured in radians, which are unitless).
- Time (t) has the dimension of [T].

**Calculation**:
\[ \text{Dimension of } \omega = \frac{\text{Dimension of angle}}{\text{Dimension of time}} = \frac{1}{T} = T^{-1} \]

### Step 2: Find the dimensions of angular acceleration (α)

**Definition**: Angular acceleration (α) is defined as the rate of change of angular speed with respect to time.

**Formula**:
\[ \alpha = \frac{\omega_2 - \omega_1}{t_2 - t_1} \]

- We already found that the dimension of angular speed (ω) is [T^{-1}].
- Time (t) has the dimension of [T].

**Calculation**:
\[ \text{Dimension of } \alpha = \frac{\text{Dimension of } \omega}{\text{Dimension of time}} = \frac{T^{-1}}{T} = T^{-2} \]

### Step 3: Find the dimensions of torque (τ)

**Definition**: Torque (τ) is defined as the product of force and the distance from the pivot point.

**Formula**:
\[ \tau = F \cdot r \]

- The dimension of force (F) is [M L T^{-2}].
- The dimension of distance (r) is [L].

**Calculation**:
\[ \text{Dimension of } \tau = \text{Dimension of } F \cdot \text{Dimension of } r = (M L T^{-2}) \cdot (L) = M L^2 T^{-2} \]

### Step 4: Find the dimensions of moment of inertia (I)

**Definition**: Moment of inertia (I) is defined as the mass times the square of the distance from the axis of rotation.

**Formula**:
\[ I = m \cdot r^2 \]

- The dimension of mass (m) is [M].
- The dimension of distance (r) is [L].

**Calculation**:
\[ \text{Dimension of } I = \text{Dimension of } m \cdot (\text{Dimension of } r)^2 = M \cdot (L^2) = M L^2 \]

### Summary of Dimensions
- **Angular speed (ω)**: \( T^{-1} \)
- **Angular acceleration (α)**: \( T^{-2} \)
- **Torque (τ)**: \( M L^2 T^{-2} \)
- **Moment of inertia (I)**: \( M L^2 \)

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