Problem 86 - Algebra and Trigonometry

Question

Bicycle Wheel The sprockets and chain of a bicycle are shown in the figure. The pedal sprocket has a radius of 4 in. the wheel sprocket a radius of 2 in. and the wheel a radius of 13 in. The cyclist pedals at 40 \(\mathrm{rpm}\) . (a) Find the angular speed of the wheel sprocket. (b) Find the speed of the bicycle. (Assume that the wheel turns at the same rate as the wheel sprocket.)

Step-by-Step Solution

Answer

(a) The angular speed of the wheel sprocket is 80 rpm. (b) The speed of the bicycle is approximately 9.075 ft/min.

Step 1: Understand the Problem

We have a bicycle with different sprocket radii and are asked to find the angular speed of the wheel sprocket and the speed of the bicycle when pedaled at 40 rpm. The given radii are: pedal sprocket 4 in, wheel sprocket 2 in, and wheel 13 in.

Step 2: Calculate Angular Speed of the Wheel Sprocket

First, we need to use the relationship between angular speeds. The chain speed must be the same for both the pedal sprocket and the wheel sprocket, so: \( \omega_1 \times r_1 = \omega_2 \times r_2 \). We know \( \omega_1 = 40 \text{ rpm} \) for the pedal sprocket with radius \( r_1 = 4 \text{ in} \), and \( r_2 = 2 \text{ in} \) for the wheel sprocket. Solving for \( \omega_2 \): \( \omega_2 = \frac{40 \times 4}{2} = 80 \text{ rpm} \). Thus, the angular speed of the wheel sprocket is 80 rpm.

Step 3: Convert Wheel Sprocket Angular Speed to Wheel Angular Speed

Since the wheel sprocket and the wheel rotate together due to being on the same axle, \( \omega_2 = \omega_{wheel} = 80 \text{ rpm} \). Therefore, the wheel also rotates at 80 rpm.

Step 4: Calculate Linear Speed of the Bicycle

To find the linear speed of the bicycle, we use the formula \( v = \omega_{wheel} \times R \). The wheel's radius \( R \) is 13 in. First, convert \( \omega_{wheel} \) from rpm to radians per minute: \( 80 \times \frac{2\pi}{60} \approx 8.377 \text{ rad/min} \). Then, \( v = 8.377 \times 13 = 108.901 \text{ in/min} \). Converting to feet per minute (1 foot = 12 inches), \( 108.901/12 \approx 9.075 \text{ ft/min} \).

Key Concepts

Bicycle Mechanics

Understanding bicycle mechanics is essential when solving problems related to bicycle motion and speed. Bicycles operate using a system of sprockets, chains, and wheels that synchronously work together when pedaled. The sprockets act like gears, and they are integral to converting the cyclist's pedaling into rotational movement at the wheel. Cyclists can change the mechanical advantage by shifting gears, allowing them to pedal more easily or accelerate faster. The way these components work together is crucial for understanding how bicycles move efficiently and quickly.

• Sprockets are like gears that transfer motion from one part of the bike to another.
• Chains connect the pedal sprocket to the wheel sprocket to enable movement.
• The size of the sprockets affects how much force is needed to pedal and how fast the bicycle can go.

With this understanding, let's delve into how they relate to angular speed and mechanical efficiency.

Angular Velocity

Angular velocity, often denoted by the symbol \(\omega\), describes how fast an object rotates. For cyclists, it measures how quickly the pedal sprocket, wheel sprocket, and wheel spin. In this exercise, we're particularly interested in the angular velocity of the wheel sprocket and how it relates to the pedal sprocket. The formula to calculate angular velocity is straightforward: \(\omega = \frac{\theta}{t}\), where \(\theta\) is the angular displacement in radians, and \(t\) is time.

In the case of this bicycle problem:

• The pedal sprocket's angular speed is given to us in revolutions per minute (rpm).
• The relationship between the pedal sprocket and the wheel sprocket is used to calculate the latter's angular velocity, indicating how efficiently torque is transferred in the system.

Ultimately, understanding this concept helps determine how a cyclist's pedaling translates into bicycle movement.

Linear Speed Calculations

Linear speed refers to how quickly the bicycle travels through space. To compute this, we use the concept that \(v = \omega_{wheel} \times R\), where \(v\) is the linear speed, \(\omega_{wheel}\) is the angular velocity of the wheel in radians per minute, and \(R\) is the wheel's radius.

In the original exercise, we must first convert the angular velocity from rpm to radians per minute to ensure accurate calculations of linear speed.

• First, convert the angular speed from rpm to radians per minute.
• Next, multiply the wheel's angular velocity by its radius to find the linear speed in inches per minute.
• Finally, convert this speed into feet per minute for more intuitive comprehension.

These steps help translate the rotational motion of the bicycle's wheel into the pace at which the bike moves forward.

Radians Per Minute

When calculating various speed measurements in bicycle mechanics, we often convert the angular velocity from revolutions per minute (rpm) to radians per minute. Radians provide a standard unit of angular measurement in physics, allowing more precise conversions and easier integration into formulas. The conversion formula you need is \(\omega_{radians} = \omega_{rpm} \times \frac{2\pi}{60}\).

The method is straightforward:

• Multiply the rpm value by \(2\pi\).
• Divide by 60 to adjust for a minute-based calculation.

This conversion is necessary because radians relate directly to the physical arc traveled, providing a more meaningful measurement in physics equations involving angular speed and linear translation.

Sprocket Mechanism

The sprocket mechanism is a fundamental component in bicycles, playing a pivotal role in transferring energy from the cyclist's feet to the bike's wheels. Understanding how this mechanism works unveils how motion is efficiently transferred and transformed.

In this context:

• The pedal sprocket connects to the cyclist's pedals, transmitting the pedaling force via a chain.
• The chain wraps tightly around both the pedal and wheel sprockets, ensuring efficient energy transfer as it rotates.
• Differently-sized sprockets change how force is converted to speed - smaller wheel sprockets increase angular velocity, making the bike go faster with less pedaling effort.

This mechanism facilitates the conversion of human energy into kinetic energy, dictating the speed and efficiency of bicycles, making it an essential topic for understanding how bicycles function.