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 determine which factors the acceleration of a body rolling down an inclined plane does not depend on, we can analyze the situation step by step.

### Step 1: Understanding Rolling Motion
Rolling motion consists of both translational and rotational motion. When a body rolls down an incline, it has both linear acceleration of its center of mass and angular acceleration about its axis.

### Step 2: Acceleration of the Center of Mass
The acceleration of the center of mass (a_cm) of a rolling body can be expressed as:
\[ a_{cm} = \frac{g \sin \theta}{1 + \frac{k^2}{r^2}} \]
where:
- \( g \) is the acceleration due to gravity,
- \( \theta \) is the angle of inclination,
- \( k \) is the radius of gyration,
- \( r \) is the radius of the rolling object.

### Step 3: Analyzing Dependencies
From the formula, we can see that:
- The acceleration \( a_{cm} \) depends on:
- The angle of inclination \( \theta \) (through \( \sin \theta \)),
- The gravitational acceleration \( g \),
- The radius of gyration \( k \) (which relates to the distribution of mass in the object).

### Step 4: Identifying Non-Dependencies
Now, we need to identify what the acceleration does not depend on:
- **Length of the Inclined Plane**: The length of the incline does not affect the acceleration of the center of mass. It only affects how long it takes to reach the bottom but not the acceleration itself.
- **Inclination Angle**: The acceleration does depend on the sine of the angle of inclination.
- **Gravitational Acceleration**: The acceleration does depend on \( g \).
- **Radius of Gyration**: The acceleration does depend on the radius of gyration.

### Conclusion
Thus, the acceleration of the body rolling down the inclined plane does not depend on the length of the incline.

### Final Answer
**The acceleration will not depend on the length of the inclined plane.**

---