A body is rolling down an inclined plane. Its acceleration will not depend on the following:

Options

A. inclination of the plane.
B. gravitational acceleration.
C. radius of gyration.
D. length of the inclined plan

Answer

To solve the problem, we need to analyze the factors that affect the acceleration of a body rolling down an inclined plane. We will break down the steps involved in understanding the relationship between the various parameters and the acceleration of the body.

### Step-by-Step Solution:

1. **Understanding the Forces Acting on the Body:**
- When a body rolls down an inclined plane, the forces acting on it include gravitational force (mg) acting downwards and the component of gravitational force along the incline (mg sin θ).
- The normal force acts perpendicular to the incline but does not affect the motion along the incline.

2. **Identifying the Acceleration:**
- The acceleration of the body can be derived from Newton's second law. The net force acting along the incline is mg sin θ, which causes the acceleration.
- The equation for acceleration (a) can be expressed as:
\[ a = \frac{mg \sin \theta}{m + \frac{I}{R^2}} \]
- Here, I is the moment of inertia of the body, and R is the radius of the body.

3. **Analyzing the Dependencies:**
- From the equation, we can see that:
- The acceleration (a) is directly proportional to \(g\) (gravitational acceleration).
- The acceleration (a) is also directly proportional to \(\sin \theta\) (the angle of inclination).
- The acceleration is dependent on the moment of inertia (I) and the radius (R) of the body.

4. **Identifying What It Does Not Depend On:**
- The key point is that the acceleration does not depend on the length of the incline. The length of the incline affects the distance traveled but does not influence the acceleration down the incline.
- Therefore, the acceleration of the body rolling down the incline is independent of the length of the path.

5. **Conclusion:**
- Based on the analysis, the acceleration of a body rolling down an inclined plane will not depend on the length of the incline. Thus, the correct answer is that the acceleration will not depend on the length of the incline.

### Final Answer:
The acceleration of a body rolling down an inclined plane will not depend on the **length of the inclined path**.

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