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Step-by-Step Solution
Step 1: Understand the Physical Situation
A solid sphere of mass $m$ and radius $R$ is placed on an inclined plane with an angle of inclination $\theta$. We compare two scenarios:
Sphere slipping down without rolling.
Sphere rolling down without slipping.
We want the ratio of the acceleration in the rolling case to the acceleration in the slipping case.
Step 2: Acceleration When Slipping (No Rolling)
When the sphere slips (pure sliding) without any rotational motion, there is no frictional force providing a torque. Hence, the net force acting along the incline is simply $mg \sin \theta$. Therefore, the acceleration in this case is:
$a_{\text{slipping}} = g \sin \theta.$
Step 3: Acceleration When Rolling Without Slipping
For a sphere rolling without slipping, a frictional force provides the necessary torque. The translational acceleration $a_{\text{rolling}}$ for a rolling object can be expressed as:
$a_{\text{rolling}} = \frac{g \sin \theta}{1 + \frac{k^2}{R^2}},$
where $k$ is the radius of gyration. For a solid sphere,
$$
\frac{k^2}{R^2} = \frac{2}{5}.
$$
Substituting this into the expression:
$a_{\text{rolling}} = \frac{g \sin \theta}{1 + \frac{2}{5}}
= \frac{g \sin \theta}{\frac{7}{5}}
= \frac{5}{7} \, g \sin \theta.$
Step 4: Find the Required Ratio of Accelerations
We now take the ratio of the rolling acceleration to the slipping acceleration:
$\frac{a_{\text{rolling}}}{a_{\text{slipping}}}
= \frac{\frac{5}{7} \, g \sin \theta}{g \sin \theta}
= \frac{5}{7}.$
Thus, the ratio of the accelerations (rolling : slipping) is:
$\boxed{\frac{5}{7}} \quad \text{or in the form } 5 : 7.
