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Step-by-Step Solution
Step 1: Identify the Scenario
We have three different objects: a solid sphere, a hollow sphere, and a ring. They are placed on a frictionless inclined plane and simply slide down without rolling.
Step 2: Determine the Forces Acting
The only forces acting on each body along the inclined plane are:
The component of gravity parallel to the incline: $mg \sin \theta$
The normal force perpendicular to the plane (which does not affect motion along the plane)
Since the plane is frictionless, there is no frictional force opposing the motion.
Step 3: Calculate the Acceleration
For any object sliding down a frictionless inclined plane, the net force along the incline is $mg \sin\theta$, and the mass is $m$, so the acceleration $a$ is given by:
$a = \frac{mg \sin \theta}{m} = g \sin \theta$
Step 4: Compare the Acceleration of All Bodies
Because none of the bodies is rolling (no rotational motion) and the plane is frictionless, each body experiences the same net force per unit mass along the plane. Hence, each object’s acceleration down the plane is the same, $g \sin\theta$.
Conclusion
All three objects have the same acceleration down the inclined plane under the given conditions. Therefore, the correct answer is that they all have the same acceleration.
