A discus thrower accelerates a discus from rest to a speed...

A discus thrower accelerates a discus from rest to a speed of $25.0 \mathrm{~m} / \mathrm{s}$ by whirling it through $1.25$ rev. Assume the discus moves on the arc of a circle $1.00 \mathrm{~m}$ in radius. (a) Calculate the final angular speed of the discus. (b) Determine the magnitude of the angular acceleration of the discus, assuming it to be constant. (c) Calculate the acceleration time.

A discus thrower accelerates a discus from rest to a speed of $25.0 \mathrm{~m} / \mathrm{s}$ by whirling it through $1.25$ rev. Assume the discus moves on the arc of a circle $1.00 \mathrm{~m}$ in
radius. (a) Calculate the final angular speed of the discus. (b) Determine the magnitude of the angular acceleration of the discus, assuming it to be constant.
(c) Calculate the acceleration time.

Key Concepts

Time Calculation in Uniformly Accelerated Rotational Motion

The time taken for an object to reach a certain angular speed under constant angular acceleration can be found using rotational kinematics equations such as ? = ?? + ?t. This concept is parallel to time calculations in linear motion under constant acceleration, making it fundamental for determining the duration of the acceleration phase in rotational motion problems.

Angular and Tangential Speed Relationship

In rotational dynamics, the tangential (linear) speed of a point on a rotating object is directly related to its angular speed by the relation v = ?r, where v is the tangential speed, ? is the angular speed, and r is the radius of the circular path. This concept allows one to convert between linear and rotational quantities, which is essential for solving problems that involve rotational kinematics.

Conversion of Revolutions to Radians

In rotational motion, angles are often measured in radians. One revolution is equivalent to 2? radians. Converting revolutions to radians is crucial for applying formulas in rotational dynamics, as these formulas require angular displacements in radians to accurately compute angular velocity, angular acceleration, and other related quantities.

Rotational Kinematics Under Constant Angular Acceleration

When an object experiences constant angular acceleration, it obeys rotational motion equations analogous to linear motion equations. For example, the equation ?² = ??² + 2?? relates the final angular speed (?), the initial angular speed (??), the angular acceleration (?), and the angular displacement (?). These equations allow one to compute unknown variables when certain initial and final conditions are provided.

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