$r$ is a variable that represents the distance from a point of reference, such as the center of rotation, to a specific point or object undergoing rotational motion. It is a fundamental parameter in the study of kinematics of rotational motion, as it determines the linear speed and acceleration of a rotating object.
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$r$ is the perpendicular distance from the axis of rotation to the point or object undergoing rotational motion.
The linear speed of a rotating object is directly proportional to its angular velocity and the distance $r$ from the axis of rotation.
The linear acceleration of a rotating object is directly proportional to its angular acceleration and the distance $r$ from the axis of rotation.
The centripetal acceleration of a rotating object is inversely proportional to the distance $r$ from the axis of rotation.
The moment of inertia of a rotating object is dependent on the distribution of mass and the distance $r$ from the axis of rotation.
Review Questions
Explain how the distance $r$ from the axis of rotation affects the linear speed of a rotating object.
The linear speed of a rotating object is directly proportional to its angular velocity and the distance $r$ from the axis of rotation. This means that as the distance $r$ increases, the linear speed of the object also increases, given the same angular velocity. This relationship is expressed by the equation $v = \omega r$, where $v$ is the linear speed, $\omega$ is the angular velocity, and $r$ is the distance from the axis of rotation.
Describe the relationship between the distance $r$ and the centripetal acceleration of a rotating object.
The centripetal acceleration of a rotating object is inversely proportional to the distance $r$ from the axis of rotation. This means that as the distance $r$ increases, the centripetal acceleration decreases, given the same angular velocity. This relationship is expressed by the equation $a_c = \frac{v^2}{r}$, where $a_c$ is the centripetal acceleration, $v$ is the linear speed, and $r$ is the distance from the axis of rotation.
Analyze how the distribution of mass and the distance $r$ from the axis of rotation affect the moment of inertia of a rotating object.
The moment of inertia of a rotating object is dependent on the distribution of mass and the distance $r$ from the axis of rotation. The moment of inertia is a measure of an object's resistance to changes in its rotational motion. Objects with more mass distributed farther from the axis of rotation will have a higher moment of inertia, making them more difficult to accelerate or decelerate. This relationship is expressed by the equation $I = \sum m_i r_i^2$, where $I$ is the moment of inertia, $m_i$ is the mass of each individual component, and $r_i$ is the distance of each component from the axis of rotation.