Set 8 Problem number 3

• Important Note on Notation

• Problem

• Solution

• Generalized Solution

• Explanation in terms of Figure(s), Extension

• Figure(s)

Problem

A circular disk of radius 4 meters is spinning through 9.99 revolutions per second.

• How many radians per second is this?
• How fast would an object attached to the rim of the disk be moving?

Solution

Each revolution is 2 `pi radians, so

• 9.99 revolutions is 19.98 `pi radians, so
• 9.99 revolutions per second is 19.98 `pi radians per second, or
• 62.7372 radians per second.

Each revolution corresponds to moving 2 `pi ( 4 meters) of circumference.

• 9.99 revolutions would be 9.99 times this, or 250.9 meters, so 9.99 revolutions per second would be 250.9 meters per second.

Alternatively, since each radian on a 4 meter circle corresponds to 4 meters of arc distance, 62.7372 radians per second corresponds to ( 62.7372 )( 4 ) meters per second = 250.9 meters per second. 

Generalized Solution

Since a revolution is 2 `pi radians, n revolutions per second corresponds to n * 2 `pi = 2 `pi * n radians per second.

• On a circle of radius r, since each radian corresponds to distance r on the circle, the velocity of a point on the circle will be 2 `pi * n * r.