A ball having a mass of 8 kg and initial speed of v1=0.2 m / s rolls over a 30 -mm-long depres

step 1 So in this question, omega 1 equal 0 .2 over 0 .125 which is equal 1 .6 radium per second and om

A ball having a mass of 8 kg and initial speed of v1=0.2 m...

A ball having a mass of $8 \mathrm{~kg}$ and initial speed of $v_{1}=0.2 \mathrm{~m} / \mathrm{s}$ rolls over a 30 -mm-long depression. Assuming that the ball rolls off the edges of contact first $A$, then $B$, without slipping, determine its final velocity $\mathbf{v}_{2}$ when it reaches the other side.

Key Concepts

Moment of Inertia

The moment of inertia is a measure of an object’s resistance to changes in its rotational motion about an axis. It depends on the object's mass distribution relative to the axis of rotation. In rolling motion problems, knowing the moment of inertia is crucial for determining the rotational kinetic energy and understanding the dynamics of the object as it rolls.

Rotational Kinetic Energy

Rotational kinetic energy is the energy due to an object's rotation and is given by (1/2)I?², where I is the moment of inertia and ? is the angular velocity. This form of energy must be taken into account along with translational kinetic energy when analyzing the motion of rolling objects.

Rolling Without Slipping

This concept refers to the condition where an object rolls in such a way that the point of contact with the surface is momentarily at rest. In problems involving rolling motion, this relationship ties the linear velocity of the center of mass to the angular velocity of the object, and is essential for relating translational and rotational motion.

Conservation of Energy

Conservation of energy is a fundamental principle stating that energy cannot be created or destroyed, only transformed. In the context of rolling motion, it ensures that the initial kinetic and potential energies are converted into translational and rotational kinetic energies, allowing for the calculation of speeds after accounting for energy transfers.

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