Two gears A and B are turning in mesh. Gear A 's radius to...

Two gears $A$ and $B$ are turning in mesh. Gear $A$ 's radius to the point of contact between the gears is $8.0 \mathrm{cm}$ and that of gear $B$ is $4.0 \mathrm{cm}$. (a) What is the linear speed of the contact point when gear $A$ 's angular velocity is $6.0 \mathrm{rad} / \mathrm{s}$ counterclockwise? (b) What is $B$ 's angular velocity?

Two gears $A$ and $B$ are turning in mesh. Gear $A$ 's radius to the point of contact between the gears is $8.0 \mathrm{cm}$ and that of gear $B$ is $4.0 \mathrm{cm}$.
(a) What is the linear speed of the contact point when gear $A$ 's angular velocity is $6.0 \mathrm{rad} / \mathrm{s}$ counterclockwise?
(b) What is $B$ 's angular velocity?

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Key Concepts

Linear Velocity at a Point

Linear velocity refers to the speed at which a point on a rotating object moves along its circular path. It is directly related to both the angular velocity and the radius of the rotation through the relation v = ?r, where v is the linear velocity, ? is the angular velocity, and r is the radius. This concept is crucial in determining the speed at the point of contact between meshing gears.

Gear Ratio

Gear ratio is the relationship between the sizes (radii or number of teeth) of two gears that are in mesh, which determines how their angular velocities are related. When two gears are meshed, the linear speed at the point of contact is the same for both, leading to the relation ??r? = ??r?. Therefore, a larger gear will rotate slower than a smaller gear if they are to maintain the same contact speed, making gear ratio an essential concept for analyzing the motion of interconnected gears.

Angular Velocity

Angular velocity is a measure of the rate of rotation of an object and is defined as the angle swept per unit time. It is a vector quantity that indicates both the speed and the direction of rotation, and it is typically measured in radians per second. In problems involving rotating bodies like gears, angular velocity is used to relate the rotational motion to other quantities such as linear speed.

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