When different regular bodies roll down along an inclined plane from rest, then acceleration will be maximum for a body whose -

Options

A. radius of gyration is least
B. mass is least
C. surface area is maximum
D. moment of inertia is maximum

Answer

To solve the problem of determining which body will have the maximum acceleration when rolling down an inclined plane, we can follow these steps:

### Step 1: Understand the Forces Acting on the Body
When a body rolls down an inclined plane, the gravitational force acting on it can be resolved into two components:
- The component acting parallel to the incline: \( F_{\parallel} = mg \sin \theta \)
- The component acting perpendicular to the incline: \( F_{\perpendicular} = mg \cos \theta \)

### Step 2: Identify the Acceleration of the Center of Mass
The acceleration of the center of mass (\( a_{CM} \)) for a rolling body can be expressed as:
\[ a_{CM} = \frac{mg \sin \theta}{m + \frac{I}{r^2}} \]
where:
- \( m \) is the mass of the body,
- \( I \) is the moment of inertia of the body,
- \( r \) is the radius of the body.

### Step 3: Substitute Moment of Inertia
The moment of inertia \( I \) can be related to the radius of gyration \( k \) by the formula:
\[ I = m k^2 \]
Substituting this into the expression for acceleration gives:
\[ a_{CM} = \frac{mg \sin \theta}{m + \frac{mk^2}{r^2}} = \frac{g \sin \theta}{1 + \frac{k^2}{r^2}} \]

### Step 4: Analyze the Expression for Acceleration
From the equation \( a_{CM} = \frac{g \sin \theta}{1 + \frac{k^2}{r^2}} \), we can see that:
- The acceleration \( a_{CM} \) is inversely proportional to \( 1 + \frac{k^2}{r^2} \).
- Therefore, to maximize \( a_{CM} \), we need to minimize \( k^2 \) (the square of the radius of gyration).

### Step 5: Evaluate the Options
Now, let's evaluate the options given in the question:
- **Option A:** Radius of gyration is least. (This is correct because a smaller \( k \) leads to maximum acceleration.)
- **Option B:** Mass is least. (Mass does not affect the acceleration in this equation.)
- **Option C:** Surface area is maximum. (Surface area does not appear in the equation.)
- **Option D:** Moment of inertia is maximum. (This would decrease acceleration, as it is in the denominator.)

### Conclusion
The body that will have the maximum acceleration when rolling down the inclined plane is the one whose radius of gyration is least. Therefore, the correct answer is **Option A**.

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