If gear A has 30 teeth and is spinning at 800rpm, gear B has 50 teeth and gear C has 20 teeth and gear D is spinning at 300rpm. How many teeth does gear D have?

rpms if all in a row

If any are on the same shaft. I do not have your picture.
B 800*(30/50)
C 800*(30/50)*(50/20)
D 800*(30/50)*(50/20)*(20/N) = 300

or more easily since all teeth are the same size
800*(30/N) = 300
N = 80

B and C are on the same shaft

Now you tell me !

B = 800*30/50 = 480 rpm

so
C = 480 rpm
D = 480 (20/N) = 300
N = 32

thank you

You are welcome :)

How exactly does Free Body Diagrams work?

Think of the body all alone in space

Add up all the force vectors on it and equate that to mass times acceleration.

To determine the number of teeth on gear D, we can use the concept of gear ratios. The gear ratio is defined as the ratio of the number of teeth on two gears.

We can set up a proportionality equation between the gears A, B, C, and D:

Gear A teeth / Gear B teeth = Gear C teeth / Gear D teeth

Substituting the given values:

30 / 50 = 20 / Gear D teeth

To solve for Gear D teeth, we can cross multiply:

30 × Gear D teeth = 50 × 20

30 × Gear D teeth = 1000

Dividing both sides of the equation by 30:

Gear D teeth = 1000 / 30 = 33.33 (approximated)

Since gears typically have whole numbers of teeth, we can round the value to the nearest integer. Therefore, gear D would have approximately 33 teeth.