A solid cylinder (i) rolls down (ii) slides down an inclined plane. The ratio of the accelerations in these conditions is

Options

A. `3:2`
B. `2:3`
C. `sqrt3:sqrt2`
D. `sqrt2:sqrt3`

Answer

To solve the problem of finding the ratio of accelerations of a solid cylinder rolling down and sliding down an inclined plane, we will follow these steps:

### Step 1: Determine the acceleration of the rolling cylinder
For a solid cylinder rolling down an inclined plane, the acceleration \( a_1 \) can be derived from the equation of motion considering both translational and rotational dynamics. The formula for the acceleration of a rolling object is given by:

\[ a_1 = \frac{g \sin \theta}{1 + \frac{I}{m r^2}} \]

Where:
- \( g \) is the acceleration due to gravity,
- \( \theta \) is the angle of inclination,
- \( I \) is the moment of inertia,
- \( m \) is the mass of the cylinder,
- \( r \) is the radius of the cylinder.

For a solid cylinder, the moment of inertia \( I \) is given by:

\[ I = \frac{1}{2} m r^2 \]

Substituting this into the acceleration formula:

\[ a_1 = \frac{g \sin \theta}{1 + \frac{\frac{1}{2} m r^2}{m r^2}} = \frac{g \sin \theta}{1 + \frac{1}{2}} = \frac{g \sin \theta}{\frac{3}{2}} = \frac{2}{3} g \sin \theta \]

### Step 2: Determine the acceleration of the sliding cylinder
For a solid cylinder sliding down an inclined plane, the only force acting along the incline is the component of gravitational force. Thus, the acceleration \( a_2 \) is given by:

\[ a_2 = g \sin \theta \]

### Step 3: Calculate the ratio of the accelerations
Now we need to find the ratio of the accelerations \( \frac{a_1}{a_2} \):

\[ \frac{a_1}{a_2} = \frac{\frac{2}{3} g \sin \theta}{g \sin \theta} \]

The \( g \sin \theta \) terms cancel out:

\[ \frac{a_1}{a_2} = \frac{2}{3} \]

### Conclusion
The ratio of the accelerations of the solid cylinder rolling down and sliding down the inclined plane is:

\[ \frac{a_1}{a_2} = \frac{2}{3} \]