10.1 Rotational Variables

Rotational motion is all about spinning objects. We'll dive into the key variables that describe how things rotate, like angular position, velocity, and acceleration. These concepts help us understand everything from spinning wheels to revolving doors.

We'll explore how rotational motion relates to linear motion and learn to calculate important values. Understanding these ideas is crucial for grasping the physics of rotating objects in our everyday world.

Rotational Motion Variables

Rotational variables in fixed-axis rotation

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• Angular position ($\theta$) measures the angle through which an object has rotated from a reference point (initial position) in radians (rad)
• Angular velocity ($\omega$) represents the rate of change of angular position with respect to time measured in radians per second (rad/s) (speedometer reading)
• Angular acceleration ($\alpha$) describes the rate of change of angular velocity with respect to time measured in radians per second squared (rad/s²) (accelerating or decelerating a spinning wheel)
• Fixed-axis rotation involves an object rotating about a fixed axis or pivot point (spinning top, revolving door) where all points on the object move in circular paths centered on the axis of rotation
• Angular displacement is the change in angular position over time

Angular velocity vs tangential speed

• Tangential speed ([object Object],[object Object]) is the linear speed of a point on a rotating object (speed of a point on the rim of a spinning wheel)
• [object Object],[object Object] relates tangential speed to angular velocity ($\omega$) and the radial distance ($r$) from the axis of rotation to the point of interest
• The direction of the tangential velocity is always perpendicular to the radius (tangent to the circular path)
• Centripetal acceleration is the acceleration of an object moving in a circular path directed toward the center of rotation

Angular velocity from position functions

• Angular velocity is the first derivative of angular position with respect to time $\omega = \frac{d\theta}{dt}$
• If the angular position is given as a function of time, $\theta(t)$ (pendulum's angle as a function of time), the angular velocity can be found by differentiating the function

Computation of rotational kinematics

• Angular velocity can be calculated using the relationship $\omega = \frac{d\theta}{dt}$ (calculating the angular velocity of a rotating fan)
• Angular acceleration is the first derivative of angular velocity with respect to time $\alpha = \frac{d\omega}{dt}$
• If the angular velocity is given as a function of time, $\omega(t)$ (angular velocity of a spinning top over time), the angular acceleration can be found by differentiating the function

Average angular acceleration calculation

• Average angular acceleration ([object Object],[object Object]) is the change in angular velocity divided by the time interval over which the change occurs [object Object],[object Object]
  • $\omega_f$ is the final angular velocity (final speed of a spinning wheel)
  • $\omega_i$ is the initial angular velocity (initial speed of a spinning wheel)
  • $t_f$ is the final time
  • $t_i$ is the initial time

Instantaneous acceleration from velocity functions

• Instantaneous angular acceleration is the first derivative of angular velocity with respect to time at a specific instant $\alpha = \frac{d\omega}{dt}$
• If the angular velocity is given as a function of time, $\omega(t)$ (angular velocity of a rotating fan blade), the instantaneous angular acceleration can be found by differentiating the function and evaluating it at the desired time (acceleration at a specific moment)

Rotational dynamics

• Moment of inertia is a measure of an object's resistance to rotational acceleration
• Torque is the rotational equivalent of force, causing angular acceleration
• Angular momentum is the rotational analog of linear momentum, describing the tendency of a rotating object to maintain its rotation