Question

Some bacteria are propelled by biological motors that spin hair-like flagella. A typical bacterial motor turning at a constant angular velocity has a radius of 1.57 x 10-8 m, and a tangential speed at the rim of 2.21 x 10-5 m/s.
(a) What is the angular speed (the magnitude of the angular velocity) of this bacterial motor?
(b) How long does it take the motor to make one revolution?

Answer 1

Answer:

(a) $\omega=1.41*10^3\frac{rad}{s}$

(b) $t=4.46*10^{-3}s$

Explanation:

The angular speed is a measure of the rotation speed. Thus, It is defined as the angle rotated by a unit of time:

$\omega=\frac{\theta}{t}(1)$

The arc length in a circle is given by:

$s=r\theta\\\theta=\frac{s}{r}(2)$

s is the length of an arc of the circle, so $v=\frac{s}{t}$.

Replacing (2) in (1):

$\omega=\frac{s}{t}\frac{1}{r}\\\omega=\frac{v}{r}$

a) Now, we calculate the angular speed:

$\omega=\frac{2.21*10^{-5}\frac{m}{s}}{1.57*10^{-8}m}\\\omega=1.41*10^3\frac{rad}{s}$

b) We use (1) to calculate the time it takes to make one revolution, which means that $\theta$ is $2\pi$.

$\omega=\frac{\theta}{t}\\t=\frac{\theta}{\omega}\\t=\frac{2\pi}{1.41*10^3\frac{rad}{s}}\\t=4.46*10^{-3}s$