Ch. 10 - Rotational Motion

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 10 - Rotational Motion

Problem 87a

Chapter 10, Problem 87a

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.

Verified step by step guidance

The problem involves understanding the relationship between the angular velocities of the front sprocket (ωբ) and the rear wheel (ωᵣ) of a bicycle. Start by noting that the chain links engage with the teeth of both sprockets, meaning the linear speed of the chain is the same at both sprockets.

The linear speed of the chain at the front sprocket is given by v = Rբ * ωբ, where Rբ is the radius of the front sprocket and ωբ is its angular velocity. Similarly, the linear speed of the chain at the rear sprocket is v = Rᵣ * ωᵣ, where Rᵣ is the radius of the rear sprocket and ωᵣ is its angular velocity.

Since the linear speed of the chain is the same at both sprockets, equate the two expressions: Rբ * ωբ = Rᵣ * ωᵣ.

The number of teeth on the sprockets (Nբ for the front and Nᵣ for the rear) is proportional to their radii because the teeth are evenly spaced. Thus, Rբ / Rᵣ = Nբ / Nᵣ.

Substitute Rբ / Rᵣ = Nբ / Nᵣ into the equation Rբ * ωբ = Rᵣ * ωᵣ to find the relationship between the angular velocities: ωᵣ = (Nբ / Nᵣ) * ωբ. This shows that the angular velocity of the rear wheel is proportional to the angular velocity of the front sprocket, scaled by the ratio of the number of teeth on the sprockets.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 pedaling translates into motion.

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 a bicycle, the gear ratio between the front and rear sprockets affects how much the rear wheel turns for each pedal rotation, influencing speed and torque.

Mechanical Advantage

Mechanical advantage refers to the factor by which a mechanism multiplies the force or torque applied to it. In bicycles, the arrangement of gears provides mechanical advantage, allowing the rider to efficiently convert pedaling force into wheel rotation. Understanding this concept helps explain how different gear combinations can optimize performance for various terrains.

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Textbook Question

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.

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Textbook Question

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 sprockets have 52 and 13 teeth, respectively.

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