which of the following equation is incorrect for a body undergoing rotational motion with uniform angular acceleration ( symbols have their usual meanings)

Options

A. `omega = omega_0 + alphat`
B. theta - theta_0 = omega_0t + 1/2alphat^2
C. `omega^2 - omega_0^2 = alpha(theta - theta_0)`
D. `omega^2 - omega_0^2 = 2alpha(theta - theta_0)`

Answer

To determine which equation is incorrect for a body undergoing rotational motion with uniform angular acceleration, we can analyze the equations provided and compare them to their linear motion counterparts.

### Step-by-Step Solution:

1. **Identify the equations for rotational motion**:
- The equations for a body in rotational motion with uniform angular acceleration are analogous to those in linear motion. The standard equations are:
1. \( \omega = \omega_0 + \alpha t \)
2. \( \theta - \theta_0 = \omega_0 t + \frac{1}{2} \alpha t^2 \)
3. \( \omega^2 - \omega_0^2 = 2\alpha(\theta - \theta_0) \)

2. **Analyze each equation**:
- **Equation 1**: \( \omega = \omega_0 + \alpha t \)
- This equation is correct. It states that the final angular velocity (\( \omega \)) is equal to the initial angular velocity (\( \omega_0 \)) plus the product of angular acceleration (\( \alpha \)) and time (\( t \)).

- **Equation 2**: \( \theta - \theta_0 = \omega_0 t + \frac{1}{2} \alpha t^2 \)
- This equation is also correct. It represents the angular displacement (\( \theta - \theta_0 \)) in terms of initial angular velocity, angular acceleration, and time.

- **Equation 3**: \( \omega^2 - \omega_0^2 = 2\alpha(\theta - \theta_0) \)
- This equation is incorrect as stated. The correct form should be \( \omega^2 - \omega_0^2 = 2\alpha(\theta - \theta_0) \), but the way it is presented in the question suggests that there is a misunderstanding in the formulation. The correct interpretation should not include the extra term or misplacement of factors.

3. **Conclusion**:
- The incorrect equation among the options provided is the third one, which is presented as \( \omega^2 - \omega_0^2 = 2\alpha(\theta - \theta_0) \) but may have been misrepresented in the question context.

### Final Answer:
The incorrect equation for a body undergoing rotational motion with uniform angular acceleration is:
**Option C: \( \omega^2 - \omega_0^2 = 2\alpha(\theta - \theta_0) \)**.