The wheel has a mass of 50 kg and a radius of gyration of 125 mm about its center of mass G .

step 1 In this question, since the wheel is without slipping, so VG1 equals omega 1R equals 5 times 0 .

The wheel has a mass of 50 kg and a radius of gyration of...

The wheel has a mass of $50 \mathrm{~kg}$ and a radius of gyration of $125 \mathrm{~mm}$ about its center of mass $G .$ If it rolls without slipping with an angular velocity of $\omega_{1}=5 \mathrm{rad} / \mathrm{s}$ before it strikes the step at $A$, determine its angular velocity after it rolls over the step. The wheel does not lose contact with the step when it strikes it.

Key Concepts

Conservation of Mechanical Energy

In the absence of energy losses (such as via friction or inelastic deformation), the total mechanical energy remains constant. For rolling objects, this involves both translational and rotational kinetic energy. This principle is useful for determining the final state of a system after a change in the motion regime, such as when the wheel negotiates a step.

Conservation of Angular Momentum about a Pivot

When a rigid body experiences an impulsive interaction at a point (such as a step contact) where the impulse passes through that point, the external torque about that point is zero. Therefore, the angular momentum about the pivot is conserved during the collision. This concept is critical for analyzing changes in rotational motion when the body’s axis of rotation shifts.

Parallel Axis Theorem

This theorem allows the calculation of an object’s moment of inertia about any axis parallel to one passing through its center of mass. It is used when the rotation axis changes, for example, when a rolling object begins to rotate about a new point of contact, by relating the moment of inertia about the center of mass to that about the new axis.

Rolling Without Slipping

This concept describes a motion where the point of contact between a rolling object and the surface is momentarily at rest relative to the surface. It means that the translational motion of the center of mass and the rotational motion about the center are linked via the relation v = ?r, which is fundamental in analyzing energy and momentum in rolling objects.

Moment of Inertia and Radius of Gyration

The moment of inertia is a measure of an object’s resistance to changes in its rotational motion about a given axis. The radius of gyration is a derived quantity that relates the distribution of the mass to the axis of rotation; it acts as a distance at which the entire mass of the body can be assumed to be concentrated to produce the same moment of inertia about the axis.

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