How do i find the average angular speed and average linear speed to this problem?

Question

Electric toothbrushes can be effective in removing dental plaque. One model consists of a head 1.10 {\rm cm} in diameter that rotates back and forth through a 70.0^\circ angle 7600 {\rm times}/{\rm minute}. The rim of the head contains a thin row of bristles

Answer 1

First, let me give you the correct answer, which I am sure that your teacher DOES NOT WANT: Zero (0).

Basically, angular velocity and velocity are both directional so going back and forth means that you are going nowhere.

What your teacher probably wants is average absolute angular velocity and average absolute velocity.

Average absolute angular velocity is simply the total amount of angle the brush sweeps through.

This is 140 degree (note the brush goes first 70 degrees forward followed by 70 degrees backwards) * 7600 /min

= 140 deg*7600/Min * 1 min/ 60 sec

= 17733 deg/sec.

The average absolute linear speed of the outside rim of the toothbrush is:

Average Angular velocity *2pi*r/360deg
r=d/2
=17333 deg/sec *pi*d/360deg
=17333 deg/sec * pi * 1.1 cm /360 deg
=170 cm/sec
=1.7 m/sec