Solid sphere rolis re dewn an when a solane making an angle...

solid sphere rolis re dewn an when a solane making an angle 0 with the horizontal, inclined plane accelcration of its centre of mass is $a$. If the same without friction, its acceleration $a$ ' will the acte slides sphere bc - (b) $\frac{5}{7} a$ (a) $\frac{7}{2} a$ (c) $\frac{7}{5} a$ (d) $\frac{5}{2} a$

solid sphere rolis re dewn an when a solane making an angle 0 with the horizontal, inclined plane accelcration of its centre of mass is $a$. If the same without friction, its acceleration $a$ ' will the acte slides sphere bc - (b) $\frac{5}{7} a$
(a) $\frac{7}{2} a$
(c) $\frac{7}{5} a$
(d) $\frac{5}{2} a$

Key Concepts

Rolling Motion

Rolling motion is the combined translational and rotational movement of a rigid body. In scenarios such as a sphere descending an incline, both the linear acceleration and angular acceleration are present, and their relationship is governed by the no-slip condition, which states that the point of contact between the sphere and the surface is momentarily at rest relative to the surface.

Static Friction

Static friction is the force that prevents the sphere from slipping when it is rolling. It provides the necessary torque to allow the sphere to rotate as it translates down the incline. This friction differs from kinetic friction, as it only acts up to the maximum value required to maintain the rolling without slipping condition.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. For a solid sphere, the moment of inertia about its center is a specific fraction of its mass and radius squared. This property is crucial in determining how the gravitational potential energy is divided between translational and rotational kinetic energy when rolling.

Acceleration on an Inclined Plane

The acceleration of an object on an inclined plane depends on the gravitational component along the plane. For a frictionless scenario, the acceleration is simply g sin?. However, when the object rolls without slipping, part of the gravitational force is used to generate rotational motion, leading to a different effective translational acceleration.

Energy Distribution in Rolling versus Sliding

When a sphere rolls without slipping, the gravitational potential energy is converted into both translational and rotational kinetic energy. In contrast, if the sphere slides without friction, the entire energy conversion is into translational kinetic energy. This difference in energy distribution leads to different accelerations in the two cases.

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