The propeller of a World War II fighter plane is 2.30 m in diameter.
(a) What is its angular velocity in radians per second if it spins at 1200 rev/min?
(b) What is the linear speed of its tip at this angular velocity if the plane is stationary on the tarmac?
(c) What is the centripetal acceleration of the propeller tip under these conditions? Calculate it in meters per second squared and convert to multiples of g.
Solution:
Part A
We are converting the angular velocity \omega = 1200\ \text{rev/min} into radians per second.
\begin{align*}
\omega = & \frac{1200\ \text{rev}}{\text{min}}\times \frac{2\pi \ \text{radian}}{1\ \text{rev}} \times \frac{1 \ \text{min}}{60 \ \text{sec}} \\ \\
\omega = & 125.6637 \ \text{radians/sec} \\ \\
\omega = & 126 \ \text{radians/sec} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}Part B
We are now solving the linear speed of the tip of the propeller by relating the angular velocity to linear velocity using the formula v = r \omega . The radius is half the diameter, so r= \frac{2.30\ \text{m}}{2} = 1.15 \ \text{m} .
\begin{align*}
v & = r \omega \\ \\
v & = \left( 1.15 \ \text{m} \right)\left( 125.6637 \ \text{radians/sec} \right) \\ \\
v & = 144.5132 \ \text{m/s} \\ \\
v & = 145 \ \text{m/s} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}
Part C
From the computed linear speed and the given radius of the propeller, we can now compute for the centripetal acceleration a_{c} using the formula
a_{c} = \frac{v^2}{r}If we substitute the given values, we have
\begin{align*}
a_{c} & = \frac{v^2}{r} \\ \\
a_{c} & = \frac{\left( 144.5132 \ \text{m/s} \right)^2}{1.15 \ \text{m}} \\ \\
a_{c} & = 18160.0565 \ \text{m/s}^2 \\ \\
a_{c} & = 1.82\times 10^{4} \ \text{m/s}^2 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}We can convert this value in multiples of g
\begin{align*}
a_{c} & = 18160.0565 \ \text{m/s}^2 \times \frac{g}{9.81 \ \text{m/s}^2} \\ \\
a_{c} & = 1851.1780 g \\ \\
a_{c} & = 1.85\times 10^{3} \ g \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}
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