The 30-kg gear A has a radius of gyration about its center...

The $30-\mathrm{kg}$ gear $A$ has a radius of gyration about its center of mass $O$ of $k_{o}=125 \mathrm{~mm}$. If the $20-\mathrm{kg}$ gear rack $B$ is subjected to a force of $P=200 \mathrm{~N}$, determine the time required for the gear to obtain an angular velocity of $20 \mathrm{rad} / \mathrm{s}$, starting from rest. The contact surface between the gear rack and the horizontal plane is smooth.

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Key Concepts

Newton's Second Law for Rotation

Newton's second law for rotation states that the net torque acting on a body is equal to the product of its moment of inertia and its angular acceleration. This principle forms the basis for analyzing the dynamics of rotational motion.

Torque

Torque is the rotational equivalent of force and is calculated as the product of a force and the perpendicular distance from the point or axis of rotation. It is the driving factor behind the angular acceleration of a system.

Moment of Inertia

The moment of inertia quantifies a rigid body's resistance to changes in its angular velocity about a specific axis, analogous to how mass governs acceleration in linear motion. It depends on the mass distribution relative to the axis of rotation.

Radius of Gyration

The radius of gyration is a measure that describes how far from the rotation axis the mass of a body is effectively concentrated. It provides a convenient way to calculate the moment of inertia as the product of the total mass and the square of this distance.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time in a rotating system. It is directly influenced by the net torque acting on the body and inversely related to its moment of inertia.

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