The assembly weighs 10 lb and has a radius of gyration kG=0.6 ft about its center of mass G . Th

step 1 In this question IG equals 0 .6 square times 10 over 32 .2 which is equal 0 .1118 and the unit i

The assembly weighs 10 lb and has a radius of gyration...

The assembly weighs $10 \mathrm{lb}$ and has a radius of gyration $k_{G}=0.6 \mathrm{ft}$ about its center of mass $G .$ The kinetic energy of the assembly is $31 \mathrm{ft} \cdot \mathrm{lb}$ when it is in the position shown. If it rolls counterclockwise on the surface without slipping, determine its linear momentum at this instant.

Key Concepts

Radius of Gyration

The radius of gyration is defined as the distance from the axis of rotation at which, if the entire mass were concentrated, the moment of inertia would be the same as that of the actual mass distribution. It simplifies the calculation of rotational inertia, especially when dealing with complex shapes or assemblies, by allowing the moment of inertia to be expressed as I = m k².

Linear Momentum

Linear momentum is the product of a body's mass and its linear velocity (p = m v) and represents the quantity of motion carried by the body. In problems involving rolling motion, determining linear momentum often involves deriving the translational speed from energy considerations, making it an important concept for connecting energy and force analyses.

Rolling Without Slipping

This concept refers to a condition where a rolling body moves in such a way that its point of contact with the surface is momentarily at rest relative to that surface. It creates a kinematic relationship between translational and rotational motion, specifically v = r?, which is crucial for correctly linking linear speed and angular velocity in rolling dynamics.

Kinetic Energy of Rolling Bodies

A rolling object possesses two components of kinetic energy: translational kinetic energy (½ m v²) due to its center of mass motion and rotational kinetic energy (½ I ?²) about its center of mass. Recognizing and combining these two forms is essential for analyzing rolling motion problems where energy is exchanged between rotation and translation.

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