Final answer:
The gymnast's reversal of spin from 19 rad/s to -19 rad/s at a torque of 519 N·m and a moment of inertia of 0.051 kg·m² takes approximately 0.00373 seconds.
Step-by-step explanation:
To determine the time required for the gymnast to exactly reverse her spin, we can use the relationship between torque, moment of inertia, and angular acceleration. Using Newton's second law for rotation, the torque τ exerted on the gymnast is related to her angular acceleration α and moment of inertia I by the formula τ = I·α. The time t needed to bring the gymnast to a halt and then to reverse her angular velocity to the same magnitude but opposite direction can be found by using the angular equivalent of the kinematic equation ω = ω₀ + α·t, where ω is the final angular velocity, ω₀ is the initial angular velocity, and α is the angular acceleration.
The gymnast's initial angular velocity is ω₀ = 19 rad/s, her moment of inertia is I = 0.051 kg·m², and the torque τ applied to stop and reverse her spin is 519 N·m.
The final angular velocity ω will be -19 rad/s when she has reversed her spin.
To find the angular acceleration α, we rearrange the formula for torque to α = τ / I.
Then, using the kinematic equation, we solve for time t required for the reversal of spin.
From α = τ / I, we get α = 519 N·m / 0.051 kg·m² = 10176.47 rad/s².
To solve for time t, we use ω = ω₀ + α·t, which gives us t = (ω - ω₀) / α.
Plugging in the values, we find t = (-19 rad/s - 19 rad/s) / 10176.47 rad/s², which equals approximately 0.00373 seconds.