Questions: An airplane circles the airport at a speed of 800 km/h. The radius of the arc is 500 m. Find the rotational speed (in rad/hr) of the aircraft.

Solution Steps

The radius of the arc is given as 500 meters. To work with consistent units, we need to convert this to kilometers:

\[ 500 \, \text{m} = 0.5 \, \text{km} \]

The circumference \( C \) of a circle is given by the formula:

\[ C = 2\pi r \]

where \( r \) is the radius. Substituting the radius in kilometers:

\[ C = 2\pi \times 0.5 = \pi \, \text{km} \]

The airplane travels at a speed of 800 km/h. To find the number of complete circles (revolutions) it makes in one hour, we divide the speed by the circumference:

\[ \text{Revolutions per hour} = \frac{800}{\pi} \]

Since one complete revolution is \(2\pi\) radians, the rotational speed in radians per hour is:

\[ \text{Rotational speed} = \frac{800}{\pi} \times 2\pi = 1600 \, \text{rad/hr} \]

Final Answer