Moment of inertia of a cylinder of mass M, length L and radius R about an axis passing through its centre and perpendicular to the axis of the cylinder is `I = M ((R^2)/4 + (L^2)/12)`. If such a cylinder is to be made for a given mass of a material, the ratio `L//R` for it to have minimum possible I is :

Options

A. `2/3`
B. `3/2`
C. `sqrt(2/3)`
D. `sqrt(3/2)`

Explanation

To find the ratio \( \frac{L}{R} \) for a cylinder with minimum moment of inertia, we will follow these steps:

### Step 1: Write the expression for the moment of inertia
The moment of inertia \( I \) of a cylinder about an axis passing through its center and perpendicular to its axis is given by:
\[ I = M \left( \frac{R^2}{4} + \frac{L^2}{12} \right) \]

### Step 2: Relate mass, density, and volume
The mass \( M \) of the cylinder can be expressed in terms of its density \( \rho \) and volume \( V \):
\[ M = \rho V \]
For a cylinder, the volume \( V \) is given by:
\[ V = \pi R^2 L \]
Thus, we have:
\[ M = \rho \pi R^2 L \]

### Step 3: Substitute for mass in the moment of inertia expression
Substituting \( M \) into the moment of inertia expression, we get:
\[ I = \rho \pi R^2 L \left( \frac{R^2}{4} + \frac{L^2}{12} \right) \]

### Step 4: Simplify the moment of inertia expression
We can rewrite the moment of inertia as:
\[ I = \rho \pi L \left( \frac{R^4}{4} + \frac{R^2 L^2}{12} \right) \]

### Step 5: Differentiate the moment of inertia with respect to \( L \)
To find the minimum moment of inertia, we differentiate \( I \) with respect to \( L \) and set the derivative equal to zero:
\[ \frac{dI}{dL} = \rho \pi \left( \frac{R^4}{4} \cdot \frac{dL}{dL} + \frac{R^2 \cdot 2L}{12} \cdot \frac{dL}{dL} \right) = 0 \]
This simplifies to:
\[ \frac{R^4}{4} + \frac{R^2 L}{6} = 0 \]

### Step 6: Solve for \( L \) in terms of \( R \)
Rearranging gives:
\[ \frac{R^4}{4} = -\frac{R^2 L}{6} \]
Multiplying through by \( -24 \) to eliminate the fractions:
\[ -6R^4 = 4R^2 L \]
Thus:
\[ L = -\frac{6R^4}{4R^2} = -\frac{3R^2}{2} \]
Since \( L \) must be positive, we take the positive ratio:
\[ \frac{L}{R} = \frac{3}{2} \]

### Step 7: Find the ratio \( \frac{L}{R} \)
To find the ratio \( \frac{L}{R} \):
\[ \frac{L}{R} = \sqrt{\frac{3}{2}} \]

### Final Answer
The ratio \( \frac{L}{R} \) for the cylinder to have minimum possible moment of inertia is:
\[ \frac{L}{R} = \sqrt{\frac{3}{2}} \]