A 2.20-kg hoop 1.20 m in diameter is rolling to the right without slipping on a horizontal floor at a steady 2.60 rad/s. (c) Find the velocity vector of each of the following points, as viewed by a person at rest on the ground: (i) the highest point on the hoop; (ii) the lowest point on the hoop; (iii) a point on the right side of the hoop, midway between the top and the bottom.

Verified step by step guidance

1. First, understand that the hoop is rolling without slipping, which means the linear velocity of the center of mass (v_cm) is related to the angular velocity (ω) by the equation v_cm = ω * r, where r is the radius of the hoop.
2. Calculate the radius of the hoop. Since the diameter is given as 1.20 m, the radius r = 1.20 m / 2 = 0.60 m.
3. Determine the linear velocity of the center of mass using the relation v_cm = ω * r. Substitute ω = 2.60 rad/s and r = 0.60 m into the equation.
4. For the highest point on the hoop, the velocity vector is the sum of the linear velocity of the center of mass and the tangential velocity due to rotation. The tangential velocity at the highest point is in the same direction as v_cm, so the total velocity is v_highest = v_cm + ω * r.
5. For the lowest point on the hoop, the velocity vector is the linear velocity of the center of mass minus the tangential velocity due to rotation, as the tangential velocity is in the opposite direction to v_cm. Thus, v_lowest = v_cm - ω * r. For the point on the right side, midway between the top and bottom, the tangential velocity is perpendicular to v_cm, so use vector addition to find the resultant velocity.

Key Concepts

Rolling Motion

Rolling motion involves both translational and rotational movement. For an object rolling without slipping, the point of contact with the surface is momentarily at rest relative to the surface. The linear velocity of the center of mass is related to the angular velocity by v = ωr, where r is the radius of the hoop.

Velocity Vector

The velocity vector describes both the magnitude and direction of velocity at a specific point. For a rolling hoop, different points on the hoop have different velocities due to the combination of translational and rotational motion. The velocity at any point can be found by considering both the linear velocity of the center and the rotational velocity around the center.

Reference Frame

A reference frame is a perspective from which motion is observed and measured. In this problem, the observer is at rest on the ground, meaning velocities are measured relative to the ground. This affects how the velocities of different points on the hoop are perceived, as the ground provides a stationary reference against which motion is compared.