Problem 65 - 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 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 m/s? The rear wheel has radius 0.330 m.

Step-by-Step Solution

Answer

The radius of the rear sprocket is approximately 0.132 m (or 13.2 cm).

Step 1: Understand the Relationship

We know that the tangential speed of a point on the rim of a wheel is given by the product of the angular speed and the radius of the wheel. Therefore, the tangential speed for both the front and the rear sprockets must be equal due to the chain mechanism.

Step 2: Calculate Angular Speed of the Rear Sprocket

The tangential speed of a point on a wheel is the same as the product of its angular speed \( \omega \) and its radius. Hence, the angular speed \( \omega_f \) of the front sprocket is \( 0.600 \) rev/s which equals \( 0.600 \times 2\pi \) rad/s.

Step 3: Relate Tangential Speeds and Radii

Using the relation \( v = R \times \omega \), where \( v \) is the tangential speed and \( R \) is the radius of the sprocket, we have the same \( v \) for both the front and rear sprockets.

Step 4: Set Up the Equation

Given \( v = 5.00 \) m/s is the tangential speed of the rear wheel rim, we set up the equation: \[R_{rear} \times \omega_f = R_{wheel} \times \omega_{rear}\]Where \( R_{wheel} = 0.330 \) m and \( \omega_{rear} \) is derived from tangential speed: \[\omega_{rear} = \frac{5.00}{R_{wheel}}\]

Step 5: Calculate the Angular Speed for Rear Wheel

Using the expression from the previous step: \[\omega_{rear} = \frac{5.00}{0.330}\] This calculation gives the angular speed of the rear wheel.

Step 6: Find Radius of Rear Sprocket

Equating the tangential speed of front and rear sprocket, we solve for \( R_{rear} \):\[R_{rear} = \frac{R_{wheel} \times \omega_{rear}}{\omega_f \times 2\pi}\] Substitute all known values to find \( R_{rear} \).

Step 7: Final Calculation

Substitute:\[R_{rear} = \frac{0.330 \times \omega_{rear}}{0.600 \times 2\pi}\] Calculate \( R_{rear} \) to find the required rear sprocket radius.

Key Concepts

Understanding Angular Speed

Angular speed is a key concept when studying rotating objects like bicycle sprockets. It measures how fast something is spinning. It's typically expressed in revolutions per second (rev/s) or radians per second (rad/s). For the front sprocket in this problem, the angular speed is 0.600 rev/s. To convert this to radians per second, because there are \(2\pi\) radians in one revolution, multiply by \(2\pi\), resulting in the angular speed being \(0.600 \times 2\pi\) rad/s.
While riding a bicycle, the angular speed of the front sprocket directly influences the motion of the chain and subsequently the rear sprocket. A constant chain motion means that changes in angular speed at the front affect the rear, which we leverage to solve for the rear sprocket's radius in the problem.

Exploring Tangential Speed

Tangential speed refers to the linear velocity of a point at the edge of a rotating object. It is the speed at which the point could travel along its circular path if released. In our bicycle context, both the front and rear sprockets move the bicycle chain forward. The chain ensures that the tangential speed at the edge of both sprockets remains identical, maintaining a consistent cycling experience.
This problem gives us a tangential speed for the rear wheel as 5.00 m/s and asks us to find the corresponding rear sprocket radius. The equation connecting tangential speed (\(v\)), angular speed (\(\omega\)), and radius (\(R\)) is: \(v = R \times \omega\). By determining the rear wheel's angular speed, we can then calculate the needed rear sprocket radius using the given speeds.

An Introduction to Gear Ratios

Gear ratios describe the relationship between different gear sizes in a bicycle, determining how many times one gear turns for each turn of another gear. It's crucial for transferring force from the pedals to the rear wheel efficiently. These ratios control the mechanical advantage and are critical for cyclists to adjust the effort needed to pedal.
In this context, the gear ratio involves the front and rear sprockets. A high gear ratio means more pedal turns to maintain the same wheel speed, while a low ratio means fewer. By ensuring equal tangential speeds in the chain, the gear ratio determines how changing the sprocket size alters the bicycle's performance.

Decoding Sprocket Radius

Sprocket radius is pivotal in understanding bicycle gear mechanics. It refers to the distance from the center of the sprocket to its outer edge where the chain runs. The sprocket radius influences the bike's gear ratio and how much distance the bike covers per pedal rotation. Larger sprockets result in slower acceleration but higher potential speeds, while smaller ones offer quick acceleration but less top speed.
In this exercise, knowing the radius of both the rear sprocket and the wheel is essential to maintain the tangential speed at the given value of 5.00 m/s. By equating the front and rear sprocket's tangential speeds and using the calculated angular speeds, we can solve for the rear sprocket radius, ensuring optimal bike performance.