91Ó°ÊÓ

Angular speed refers to the rate at which an object moves through an angle. For objects moving in a circular path, like a Ferris wheel, it is essential in understanding how quickly they complete a rotation. Typically represented by the symbol \(\omega\), angular speed is measured in radians per second (rad/s).
To calculate the angular speed, we use the formula:

• \(\omega = \frac{2\pi}{T}\)

where \(T\) is the time taken for one complete rotation.
In the context of the Ferris wheel problem, it takes 12 seconds to complete a single revolution. Hence, the angular speed of the Ferris wheel is \(\frac{\pi}{6}\,\text{rad/s}\). This ensures that the wheel covers the angle of \(2\pi\) radians (one full circle) every 12 seconds.

Angular acceleration contrasts angular speed by identifying how quickly the angular speed itself changes over time. Denoted by \(\alpha\), it measures the rate of change of angular velocity and is expressed in radians per second squared (rad/s²).
When the Ferris wheel comes to a stop, its angular speed changes from an initial value to zero. This change over a specified angle (one quarter of a revolution, or \(\frac{\pi}{2}\,\text{radians}\)) allows us to calculate the angular acceleration using the following relationship:

• \(\omega_f^2 = \omega_i^2 + 2\alpha\theta\)

Here, \(\omega_i = \frac{\pi}{6}\,\text{rad/s}\) is the initial angular speed, \(\omega_f = 0\), and \(\theta = \frac{\pi}{2}\).
Through calculation, the wheel's angular acceleration is found to be \(- \frac{\pi}{72} \, \text{rad/s}^2\). The negative sign reflects deceleration, as the wheel slows to a halt.

Tangential acceleration is closely related to angular motion but describes the linear aspect. It represents how quickly the linear speed of a point on a rotating object changes as it moves along its circular path. This acceleration affects the object's tangential velocity.
Tangential acceleration, \(a_t\), can be determined from angular acceleration \(\alpha\) and the radius \(r\) of the circular path using the formula:

• \(a_t = \alpha \times r\)

With the Ferris wheel's radius of 9 meters, and angular acceleration previously calculated as \(-\frac{\pi}{72}\,\text{rad/s}^2\), we find the tangential acceleration:

• \(a_t = -\frac{\pi}{72} \times 9\,m = -\frac{9\pi}{72} \text{ m/s²}\)

The negative result indicates that the acceleration is opposite to the motion's initial direction, aligning with the wheel's slowing down. Understanding this helps in comprehending the physical experience of slowing motion on a Ferris wheel and similar rotating systems.