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. 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.

검증된 단계별 안내

First, understand that the hoop is rolling without slipping, which means the linear velocity of the center of mass is equal to the tangential velocity at the rim due to rotation. The linear velocity \( v \) of the center of mass can be calculated using \( v = r \cdot \omega \), where \( r \) is the radius of the hoop and \( \omega \) is the angular velocity.

Calculate the radius \( r \) of the hoop. Since the diameter is given as 1.20 m, the radius \( r \) is half of the diameter: \( r = \frac{1.20}{2} \) m.

Determine the linear velocity \( v \) of the center of mass using the formula \( v = r \cdot \omega \). Substitute the values: \( v = \frac{1.20}{2} \cdot 2.60 \) rad/s.

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 the linear velocity, so the total velocity is \( v_{highest} = v + r \cdot \omega \).

For the lowest point on the hoop, the tangential velocity due to rotation is in the opposite direction to the linear velocity. Therefore, the velocity vector at the lowest point is \( v_{lowest} = v - r \cdot \omega \). For the point on the right side, midway between the top and bottom, the tangential velocity is perpendicular to the linear velocity, so use vector addition to find \( v_{midway} = \sqrt{v^2 + (r \cdot \omega)^2} \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Rolling Without Slipping

Rolling without slipping is a condition where an object rolls on a surface without any relative motion between the point of contact and the surface. This means the linear velocity of the center of mass is equal to the angular velocity times the radius. For a hoop, this ensures that the translational and rotational motions are synchronized, crucial for analyzing the velocity of different points on the hoop.

Velocity of Points on a Rotating Object

The velocity of a point on a rotating object is determined by both its translational motion and its rotational motion. For a hoop rolling without slipping, the velocity at any point is the vector sum of the translational velocity of the center of mass and the tangential velocity due to rotation. This concept helps in calculating the velocity vectors for specific points on the hoop.

Reference Frames

A reference frame is a perspective from which motion is observed and measured. In this problem, the observer is at rest on the ground, which means the velocities of points on the hoop are measured relative to the ground. Understanding reference frames is essential to correctly interpret the velocities of the points on the hoop as they appear to a stationary observer.