Problem 33 - University Physics with Modern Physics

Question

While riding a multispeed bicycle, the rider can select the radius of the rear sprocket that is fixed to the rear axle. The front sprocket of a bicycle has radius 12.0 \(\mathrm{cm}\) . If the angular speed of the front sprocket is 0.600 rev/s, what is the radius of the rear sprocket for which the tangential speed of a point on the rim of the rear wheel will be 5.00 \(\mathrm{m} / \mathrm{s} ?\) The rear wheel has radius 0.330 \(\mathrm{m} .\)

Step-by-Step Solution

Answer

The radius of the rear sprocket should be approximately 11.05 m.

Step 1: Understand the Relationships

To solve this problem, understand that the tangential speeds of the rear and front sprockets are related to their angular speeds and radii. Since both sprockets are connected by a chain, they will have the same linear speed.

Step 2: Convert Angular Speed to Tangential Speed for Front Sprocket

Calculate the tangential speed of the front sprocket using the formula: \[ v = \omega \times r_{\text{front}} \]where \( \omega = 0.600 \) rev/s and needs to be converted to rad/s (\( 1 \text{ rev/s} = 2\pi \text{ rad/s} \)). Thus, \( \omega = 0.600 \times 2\pi = 1.2\pi \) rad/s. The radius of the front sprocket, \( r_{\text{front}} \), is 12.0 cm or 0.120 m. Hence:\[ v_{\text{front}} = 1.2\pi \times 0.120 = 0.144\pi \text{ m/s} \]

Step 3: Equalize Linear Velocities of Chain on Both Sprockets

Since the chain links moving over both sprockets have the same linear velocity, equate this to both the tangential speed from the front sprocket and desired speed provided for the point on the rear wheel:\[ r_{\text{rear}} \times 0.600 \times 2\pi = 5 \text{ m/s} \]

Step 4: Solve for the Rear Sprocket Radius

We need to solve:\[ r_{\text{rear}} \times 0.144\pi = 5 \]This simplifies to:\[ r_{\text{rear}} = \frac{5}{0.144\pi} \approx 11.05 \text{ m} \]

Step 5: Validate Against Given Conditions

Re-calculate using the desired condition. Recognizing that the rear wheel tangential speed is equal:The wheel radius of 0.330 m verifies desired 5 m/s if correct \( r_{\text{rear}} \) exploitable in step 4 is valid, since connected speeds via mechanisms conform via circumference at adhesive limits formulaically.

Key Concepts

Angular Speed

Angular speed is a fundamental concept in rotational dynamics. It represents how fast an object rotates around an axis. Specifically, angular speed gives us the number of revolutions per unit time. This is an important measurement for anything that spins, like wheels and gears.
In the given exercise, the angular speed of the bicycle's front sprocket is given as 0.600 revolutions per second. However, for calculations involving radians (a more universal unit of angle), it’s often useful to convert this to radians per second.
The conversion uses the fact that one revolution is equal to \(2\pi\) radians. Thus, by multiplying 0.600 by \(2\pi\), we convert the angular speed to radians per second, yielding \(1.2\pi\) rad/s. This measure helps us calculate the tangential speed of the sprocket.

Tangential Speed

Tangential speed deals with how fast a point on the edge of a rotating object moves in a linear path. It is directly related to the angular speed and the radius of the rotation. You can think of it as the speed experienced if you were to place your finger on the outer rim of a spinning plate.
For the front sprocket of a bicycle, its tangential speed can be calculated using the formula: \[ v = \omega \times r \] where \(v\) is the tangential speed, \(\omega\) is the angular speed, and \(r\) is the radius.
In the example, converting \(1.2\pi\) rad/s and multiplying by 0.120 m gives the front sprocket's tangential speed as \(0.144\pi\) m/s. This principle ensures that the chain maintains a consistent speed as it moves between the sprockets, crucial for efficient bicycle operation.

Bicycle Mechanics

Bicycle mechanics involve various systems working together to enable smooth and efficient movement. Key components include gears, chains, and sprockets, which convert the rider's effort into forward motion.
When you pedal a bike, you use the front sprocket to drive the rear sprocket via the chain. This is a brilliant mechanical design that capitalizes on principles of physics to amplify human power.
The effective functioning of these components relies on concepts like angular and tangential speeds for optimal performance. Adjusting the rear sprocket's radius can impact the bike's ease of pedaling and how fast it travels, hence why multispeed bicycles provide adjustable gears.

Chain and Sprocket System

The chain and sprocket system is what transfers energy from the pedals of a bicycle to its wheels. This classic mechanism forms the core of bicycle motion, connecting the front and rear sprockets.
The chain mimics a conveyor belt, handling the transformation of rotational motion from the front to the rear part of the bike. An essential point is that the chain's linear speed remains constant between sprockets.
In practice, selecting different sprockets (especially on multispeed bicycles) allows the rider to modify the bike's speed and ease of pedaling. By changing the radius of the rear sprocket, as seen in the exercise, riders can fine-tune their bicycle's performance to navigate varying terrains, like uphill or flat roads. This adaptability is a testament to the ingenuity of the chain and sprocket system.