Rotational Motion in Everyday Objects

This section explores rotational motion in common objects like watch hands and discus throwing.

Example: Calculating the angular velocity and tip speed of a 1.00 cm long second hand on a watch.

This problem illustrates the application of rotational kinematics to a familiar object, emphasizing the relationship between angular velocity and linear speed at different radii.

Example: A pulley system with a 1.5 kg block connected by a rope across a 50 cm diameter, 2.0 kg pulley. The problem involves determining rope tension given the block's upward acceleration.

This example combines concepts of rotational motion with force analysis, demonstrating the interplay between linear and angular dynamics.

Example: Analyzing the motion of a discus throw, where the thrower completes one revolution in 1.0 second with a constant angular acceleration.

Key calculations:

• Angular acceleration of the thrower
• Final angular velocity
• Linear speed of the discus at release

Highlight: This problem showcases the application of rotational kinematics to sports, illustrating how angular motion translates to the high linear speeds achieved in discus throwing.

These examples reinforce the practical applications of rotational motion concepts in various real-world scenarios, from simple machines to athletic performances.

Angular Motion Examples

This section covers several example problems related to angular motion, including turntable rotation, softball pitching mechanics, and Ferris wheel motion.

Example: A turntable rotating counterclockwise at 78 rpm with a speck of dust at an initial angle of 0.45 rad. After 8 seconds, the dust speck's new angular position is calculated to be 2.95 rad.

Highlight: The problem demonstrates how to convert between revolutions per minute and radians per second, as well as calculating angular displacement over time.

Vocabulary: Angular velocity (ω) - the rate of change of angular position with respect to time, measured in radians per second.

The softball windmill pitch example explores the angular acceleration and tangential acceleration of a pitcher's arm during the motion.

Example: In a softball windmill pitch, the pitcher's arm rotates through just over half a circle in 0.15 seconds, with increasing angular velocity.

Key calculations include:

• Angular acceleration of the arm
• Tangential acceleration of the ball (0.60 m from shoulder)
• Total angle of arm rotation

This problem highlights the application of rotational kinematics to sports biomechanics.