The final angular speed of the discus is 26.25 rad/s and the magnitude of the angular acceleration is 87.4 rad/s².
The student asks to calculate the final angular speed of a discus and the magnitude of the angular acceleration given that a discus thrower accelerates a discus from rest to a speed of 25.2 m/s by whirling it through 1.26 revolutions on the arc of a circle 0.96 m in radius.
Part (a) - Calculating the Final Angular Speed
To calculate the final angular speed (angular velocity), we use the relationship between linear speed (v) and angular speed (ω) which is v = rω, where r is the radius of the circle. Rearranging the formula, we have ω = v/r.
Substituting the given values:
ω = 25.2 m/s / 0.96 m = 26.25 rad/s.
Part (b) - Determining the Magnitude of the Angular Acceleration
To find the angular acceleration (α), we use the formula α = (ω - ω₀)/t, where ω₀ is the initial angular speed and t is the time taken to reach the final speed. Since the discus starts from rest, ω₀ = 0. However, we are not given the time directly. Instead, we can use the number of revolutions (n) to find the angular distance (θ) and then use kinematic equations for rotational motion.
Firstly, we convert revolutions to radians. θ = n × 2π = 1.26 × 2π rad.
With the final angular speed (ω) and angular distance (θ), we can use the equation ω² = ω₀² + 2αθ to find α.
α = (ω² - ω₀²)/(2θ). Substituting the values, we get:
α = (26.25 rad/s)² / (2 × 1.26 × 2π) = 87.4 rad/s².