Final answer:
To find the final angular velocity of the top when the string is unwound, calculate the number of turns the string makes around the top, determine the angular displacement in radians, and then use the kinematic equation for angular motion that includes the given angular acceleration.
Step-by-step explanation:
To determine the final angular velocity of the top after the string is completely unwound, we can apply the kinematic equation that relates angular velocity, angular acceleration, and angular displacement. The question informs us that the top is initially at rest, which means the initial angular velocity (ω₀) is 0 rad/s, and the angular acceleration (α) is 12 rad/s². We also know the radius (r) around which the string is wound is 2.0 cm and the total length (L) of the string is 64 cm.
First, we need to convert the radius to meters (r = 0.020 m), then calculate the number of turns (n) the string makes around the top by dividing the string's length by the circumference of the circle formed at the radius:
n = L / (2πr)
Next, we calculate the angular displacement (θ) in radians, which is the number of turns multiplied by 2π (since there are 2π radians in one full turn):
θ = 2πn
To find the final angular velocity (ω), we use the kinematic equation for angular motion:
ω² = ω₀² + 2αθ
Since ω₀ = 0, we simplify to:
ω = √(2αθ)
After substituting our values for α and θ into the equation, we calculate ω, which gives us the final angular velocity of the top after the string has been completely unwound.