Final answer:
To calculate the final angular velocity of the top, you find the total angular displacement by dividing the length of the string by the circumference at the point of wrapping, then multiplying by 2π. Using the kinematic equation for rotation (ω² = ω₀² + 2αΘ) with an angular acceleration of +15 rad/s² and solving for ω, you get approximately 31.7 rad/s.
Step-by-step explanation:
Calculating The Final Angular Velocity of A Spinning Top
To calculate the final angular velocity of the top, we must first find the number of revolutions the top makes as the string unwinds. Since the string is 74 cm long and it is wrapped around a place with a radius of 2.2 cm, we apply the formula for the circumference of a circle (C = 2πr) to find the length of the string that is wrapped around in one revolution. Then we divide the total length of the string by this number to find the total number of revolutions:
Length of one revolution = 2π(2.2 cm) = 13.82 cm
Number of revolutions = 74 cm / 13.82 cm ≈ 5.35 revolutions
The total angular displacement in radians is the number of revolutions times 2π which equals approximately 33.6 radians.
Since the angular acceleration is given as +15 rad/s², we can use the kinematic equation for rotation (ω² = ω₀² + 2αΘ) to solve for the final angular velocity ω, where ω₀ (the initial angular velocity) is zero. Therefore, the final angular velocity ω is:
Final angular velocity (ω) = √(0 + 2 * 15 rad/s² * 33.6 rad) ≈ √(1008 rad²/s²) = 31.7 rad/s
The final angular velocity of the top when the string is completely unwound is approximately 31.7 radians per second.