Question

A solid cylinder, a solid sphere, a disc and a ring are released from the top of an inclined plane (frictionless) so that they slide down the plane without rolling. The maximum acceleration down the plane is

• A. for the disc
• B. for the solid cylinder
• C. for the solid sphere
• D. for the ring
• E. the same for all

Hint:

If no rolling, rotational inertia does not matter.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This is a problem about the motion of objects on an inclined plane. The crucial information here is that the plane is frictionless and the objects slide without rolling.
Step 2: Key Formula or Approach:
When an object slides down a frictionless inclined plane, the only force acting on it along the plane is the component of gravity parallel to the incline. The acceleration is determined by Newton's Second Law. Let \(\theta\) be the angle of inclination of the plane. The force of gravity is \(mg\), acting vertically downwards. The component of gravity parallel to the plane is \(mg \sin\theta\). According to Newton's Second Law, \(F_{net} = ma\).
Step 3: Detailed Explanation:
The problem specifies that the plane is frictionless and the objects slide, not roll. This means we do not need to consider rotational motion, moments of inertia, or the shapes and mass distributions of the objects. For any object of mass `m` on the inclined plane, the net force along the plane is: \[ F_{net} = mg \sin\theta \] Applying Newton's Second Law: \[ ma = mg \sin\theta \] The mass `m` cancels out from both sides: \[ a = g \sin\theta \] This result shows that the acceleration of an object sliding down a frictionless inclined plane depends only on the acceleration due to gravity `g` and the angle of inclination \(\theta\). It is independent of the object's mass, shape, size, or moment of inertia. Since all the objects (solid cylinder, solid sphere, disc, and ring) are released on the same inclined plane, they will all experience the same acceleration.
Step 4: Final Answer:
The acceleration is the same for all objects.