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

(a)
`\omega=1.41*10^3(rad)/(s)`

(b)
`t=4.46*10^(-3)s`

Step-by-step 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=(\theta)/(t)(1)`

The arc length in a circle is given by:

`s=r\theta\\\theta=(s)/(r)(2)`

s is the length of an arc of the circle, so
`v=(s)/(t)`.

Replacing (2) in (1):

`\omega=(s)/(t)(1)/(r)\\\omega=(v)/(r)`

a) Now, we calculate the angular speed:

`\omega=(2.21*10^(-5)(m)/(s))/(1.57*10^(-8)m)\\\omega=1.41*10^3(rad)/(s)`

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

`\omega=(\theta)/(t)\\t=(\theta)/(\omega)\\t=(2\pi)/(1.41*10^3(rad)/(s))\\t=4.46*10^(-3)s`