In a two-sprocket chain mechanism, if the larger sprocket has 50 teeth and rotates at 50 rpm, how many teeth must the smaller sprocket have to achieve 150 rpm?

To determine how many teeth the smaller sprocket must have to achieve an output speed of 150 rpm when the larger sprocket is rotating at 50 rpm, we can use the relationship between the speed of rotation and the number of teeth in a chain mechanism. This relationship is based on the principle that the product of the speed (rpm) and the number of teeth for both sprockets remains constant.

Let's denote the number of teeth on the larger sprocket as ( T_L = 50 ) teeth, the rotation speed of the larger sprocket as ( N_L = 50 ) rpm, the unknown number of teeth on the smaller sprocket as ( T_S ), and its rotation speed as ( N_S = 150 ) rpm.

According to the formula:

[ N_L \times T_L = N_S \times T_S ]

Substituting the known values:

[ 50 \times 50 = 150 \times T_S ]

This simplifies to:

[ 2500 = 150 \times T_S ]

To find ( T_S ), divide both sides by 150:

[ T_S = \frac{2500}{150} ]

Calculating that gives:

[ T_S