A 20-tooth pinion drives a 40-tooth gear. The pinion has an angular velocity of 1000 rpm. We need to determine the velocity ratio and the angular velocity of the gear.

1. Problem Description

A 20-tooth pinion drives a 40-tooth gear. The pinion has an angular velocity of 1000 rpm. We need to determine the velocity ratio and the angular velocity of the gear.

2. Solution Steps

The velocity ratio is the ratio of the number of teeth on the gear to the number of teeth on the pinion.

$Velocity\,Ratio = \frac{N_{gear}}{N_{pinion}}$

where $N_{gear}$ is the number of teeth on the gear and $N_{pinion}$ is the number of teeth on the pinion.

Given that $N_{gear} = 40$ and $N_{pinion} = 20$,

$Velocity\,Ratio = \frac{40}{20} = 2$

The angular velocity of the gear can be calculated using the following relationship:

$\frac{N_{gear}}{N_{pinion}} = \frac{\omega_{pinion}}{\omega_{gear}}$

where $\omega_{pinion}$ is the angular velocity of the pinion and $\omega_{gear}$ is the angular velocity of the gear.

Given that $\omega_{pinion} = 1000 \, rpm$, we can solve for $\omega_{gear}$:

$\omega_{gear} = \frac{N_{pinion}}{N_{gear}} \times \omega_{pinion}$

$\omega_{gear} = \frac{20}{40} \times 1000\, rpm$

$\omega_{gear} = \frac{1}{2} \times 1000\, rpm$

$\omega_{gear} = 500\, rpm$

3. Final Answer

Velocity Ratio = 2

Angular velocity of the gear = 500 rpm