A fairground ride spins its occupants inside a flying saucer-shaped container. If the horizontal circular path the riders follow has an 8.00 m radius, at how many revolutions per minute will the riders be subjected to a centripetal acceleration whose magnitude is 1.50 times that due to gravity?
Solution:
Centripetal acceleration a_{c} is the acceleration experienced while in uniform circular motion. It always points toward the center of rotation. The relationship between the centripetal acceleration a_{c} and the angular velocity \omega is given by the formula
a_{c}=r\omega^{2}
Now, taking the formula and solving for the angular velocity:
\omega = \sqrt{\frac{a_{c}}{r}}
From the given problem, we are given the following values: r=8.00\ \text{m} and a_{c}=1.50\times 9.81 \ \text{m/s}^2=14.715\ \text{m/s}^2. If we substitute these values in the formula, we can solve for the angular velocity.
\begin{align*} \omega & = \sqrt{\frac{a_{c}}{r}} \\ \\ \omega & = \sqrt{\frac{14.715\ \text{m/s}^2}{8.00\ \text{m}}} \\ \\ \omega & = 1.3561\ \text{rad/sec} \\ \\ \end{align*}
Then, we can convert this value into its corresponding value at the unit of revolutions per minute.
\begin{align*} \omega & = 1.3561\ \frac{\text{rad}}{\text{sec}} \times \frac{60\ \text{sec}}{1\ \text{min}}\times \frac{1\ \text{rev}}{2\pi \ \text{rad}} \\ \\ \omega & = 12.9498\ \text{rev/min} \\ \\ \omega & = 13.0 \ \text{rev/min} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}
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