91Ó°ÊÓ

Angular acceleration is the rate of change of angular velocity with respect to time. It tells us how quickly the speed of rotation is changing.
In the given problem, the top receives an angular acceleration noted as \( +12\, \text{rad/s}^2 \). The positive sign denotes that the top is speeding up, which means it starts from rest and spins faster as the string unwinds.
Angular acceleration \( \alpha \) is expressed mathematically as:

• \( \alpha = \frac{d\omega}{dt} \)

Where \( d\omega \) is the change in angular velocity and \( dt \) is the change in time. In simpler terms, it's the difference in how fast something is spinning from one moment to the next, per unit of time.
Understanding angular acceleration is key to predicting how quickly something will reach a desired spin speed, which links directly to our exercise where the final angular velocity is sought after.

Angular displacement is the measure of the angle through which an object moves on a circular path. It represents how far the top has spun in radians, a unit that describes the angle as a fraction of the circle's radius.
In this exercise, when the string is completely unwound, the angular displacement \( \theta \) is calculated as the ratio of the string's length to the radius of the top:

• \( \theta = \frac{s}{r} \)
• \( \theta = \frac{0.64 \text{ m}}{0.02 \text{ m}} = 32 \) radians

This formula shows the total rotation in radians completed by the top. The calculation tells us how many full spins or fractions of spins the top makes as the string is pulled out. Angular displacement provides a direct way to relate the linear movement (the string unwraps) to a rotary effect (the top spins), enabling us to consider how many spins are completed before the string fully unwinds.

Angular velocity is the rate at which an object rotates or spins around a circular path. It represents how fast the angular displacement is changing over time.
For the top, when the string is unwound entirely, we need to find its final angular velocity \( \omega \). Using the kinematic equation for rotational motion:

• \( \omega^2 = \omega_0^2 + 2\alpha\theta \)
• Starting speed \( \omega_0 = 0 \), \( \alpha = 12 \text{ rad/s}^2 \), \( \theta = 32 \text{ radians} \)
• \( \omega^2 = 0 + 2 \times 12 \times 32 = 768 \)
• \( \omega = \sqrt{768} \approx 27.7 \text{ rad/s} \)

The final angular velocity is \( 27.7 \text{ rad/s} \), indicating how quickly the top is spinning now that the string has been completely unrolled. This measure can help predict the performance characteristics of any rotating system, revealing how intense the motion is after experiencing angular acceleration.