Question 797378: Inside a clock are many different gears. Suppose that one gear, which has a radius of two centimeters, drives another gear, which has a radius of seven centimeters. If the angular speed of the gear with a radius of two centimeters is 5pi/7 radians per second, what is the linear speed of one of the teeth on the gear with seven centimeter radius? (Note: Assume the teeth on the seven centimeter gear are seven centimeters from the center of the gear.) if you can help me with this that would be great. Ps... its due friday.
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! That note is not very helpful (or realistic).
What part of the teeth is 7cm from the center of the large gear? The tips of the teeth? The valleys in between the teeth? The middle of the teeth?
What about the smaller gear? Where do you measure the 2cm radius there?
What clearance do you have between the gears? Can we assume that the gaps between the tips of the teeth in one gear and the part of the other gear that meshes with them is so small that we can call it zero?
It would have been more helpful to say that the radius is measured to mid-height of the teeth on each gear, that the teeth are the same height on both, and that the gears fit very tightly, so that the gap between the parts that almost touch is so small as to be disregarded.
Let's assume that the gears'centers are  apart, and that a point at 2 cm from the center of the small gear contacts a point at 7 cm from the center of the large gear, and somehow they mesh in such a way that the gears don't slip. Maybe the teeth are so small that they are nothing compared to the radius of the gears.
Let's calculate the linear velocity of point on the large gear at 7 cm from the center, which is the same linear velocity as for the point in the small gear at 2 cm from the center.
The linear velocity of a point on the small gear is
During one second, the small gear turns  , so that each point at 2cm from the center travels along an arc of , which has a length of
 
So the points at 2cm from the center travel at  cm/s, and that is the linear speed for
points in the small gear at 2 cm from the center, and for
points in the large gear at 7 cm from the center.
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