Bicycle gears:How is the angular velocity ωᵣ of the rear wheel of a bicycle related to the angular velocity ωբ of the front sprocket and pedals? Let Nբ and Nᵣ be the number of teeth on the front and rear sprockets, respectively, Fig. 10–71, and Rբ and Rᵣ their respective radii. The teeth are spaced the same on both sprockets and the rear sprocket is firmly attached to the rear wheel. Evaluate the ratio ωᵣ / ωբ when the front and rear sprocketshave 42 and 28 teeth.

Verified step by step guidance

1. Understand the relationship between the angular velocities of the front sprocket (ωբ) and the rear wheel (ωᵣ). The chain transfers motion from the front sprocket to the rear sprocket, so the linear speed of the chain is the same for both sprockets.
2. Express the linear speed of the chain in terms of the angular velocity and radius of the sprockets. For the front sprocket, the linear speed is v = Rբ * ωբ, and for the rear sprocket, it is v = Rᵣ * ωᵣ.
3. Since the linear speed of the chain is the same for both sprockets, set the two expressions equal: Rբ * ωբ = Rᵣ * ωᵣ.
4. Relate the radii of the sprockets to the number of teeth. The radius of a sprocket is proportional to the number of teeth, so Rբ / Rᵣ = Nբ / Nᵣ. Substitute this into the equation: (Nբ / Nᵣ) * ωբ = ωᵣ.
5. Rearrange the equation to find the ratio of angular velocities: ωᵣ / ωբ = Nբ / Nᵣ. Substitute the given values for the number of teeth (Nբ = 42, Nᵣ = 28) to evaluate the ratio.

Key Concepts

Angular Velocity

Angular velocity (ω) is a measure of how quickly an object rotates around an axis, expressed in radians per second. In the context of a bicycle, it describes the rate at which the pedals and wheels turn. The relationship between the angular velocities of different components, such as the front sprocket and rear wheel, is crucial for understanding how gear ratios affect speed and torque.

Gear Ratio

The gear ratio is the ratio of the number of teeth on two gears that are meshed together. It determines how the angular velocities of the connected gears relate to each other. For bicycles, the gear ratio influences how much the rear wheel turns in relation to the pedals, affecting the bike's speed and the effort required to pedal.

Relationship Between Angular Velocities

The relationship between the angular velocities of the front sprocket (ωᵇ) and the rear wheel (ωᵣ) can be expressed using the gear ratio, which is derived from the number of teeth on the sprockets. Specifically, the ratio ωᵣ / ωᵇ can be calculated as the inverse of the gear ratio, which is determined by the number of teeth on the front and rear sprockets. This relationship is essential for evaluating how changes in gear affect the bicycle's performance.