Problem 86 - Precalculus Mathematics for Calculus

Question

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 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.) (IMAGE CAN NOT COPY)

Step-by-Step Solution

Answer

(a) 80 rpm. (b) 1.03 mph.

Step 1: Translate Pedal Speed into Sprocket Speed

To begin, identify the connection between the pedal sprocket and the wheel sprocket using their respective radii. The cyclist pedals at 40 rpm with a radius of the pedal sprocket at 4 inches. Since the chain moves through both sprockets without slipping, the linear speed at the edge of both sprockets must be equal. So, the linear speed at the pedal sprocket is \( v = r_{pedal} \times \omega_{pedal} \), where \( r_{pedal} = 4 \) inches and \( \omega_{pedal} = 40 \) rpm. Convert \( \omega_{pedal} \) from rpm to radians per minute: \[ \omega_{pedal} = 40 \times \frac{2\pi}{60} \text{ rad/min} \] We are thus attempting to solve for \( \omega_{wheel} \), with \( \omega_{wheel} \) being the angular speed of the wheel sprocket.

Step 2: Balance Linear Velocities of Sprockets

Using the connection that the linear speed \( v \) is the same for both sprockets due to the chain, we have:\[ v = r_{pedal} \times \omega_{pedal} = r_{wheel} \times \omega_{wheel} \] where \( r_{wheel} = 2 \) inches, Consequently:\[ \omega_{wheel} = \frac{r_{pedal}}{r_{wheel}} \times \omega_{pedal} = \frac{4}{2} \times 40 = 80 \text{ rpm} \] The wheel sprocket's angular speed is thus 80 rpm.

Step 3: Convert Wheel Sprocket Speed to Bicycle Speed

With the wheel sprocket turning at the same rate as the wheel itself, the next step is to find the bicycle speed using the radius of the wheel, which is 13 inches. The linear speed of the wheel (and thus the bicycle) is given by:\[ v = r_{wheel} \times \omega_{wheel} \] Substitute \( r_{wheel} = 13 \) inches and convert \( \omega_{wheel} \) to rad/min:\[ v = 13 \times \left( \frac{80 \times 2\pi}{60} \right) \text{ inches/min} \] This evaluates to \( v \approx 1,088 \) inches/min. To convert to a more standard measure like feet per minute:\[ v_{feet/min} = \frac{1,088}{12} \approx 90.67 \text{ feet/min} \] Finally, convert feet/min to miles per hour by using the conversion factor where 1 mile = 5,280 feet and 1 hour = 60 minutes:\[ v_{mph} = \frac{90.67 \times 60}{5,280} \approx 1.03 \text{ mph} \] The speed of the bicycle is therefore approximately 1.03 mph.

Key Concepts

Angular Speed

Angular speed is a measure of how quickly an object rotates or revolves relative to another point. It is typically measured in radians per unit of time, such as radians per minute or radians per second. In this exercise, the angular speed of the bicycle components is crucial in understanding how fast the wheel turns compared to the pedals.

To find the angular speed, use the formula:

• \( \omega = 2\pi \times \text{rpm} / 60 \)

This converts revolutions per minute (rpm) to radians per minute. Here, the cyclist pedals at 40 rpm, which needs converting into radians/minute to find the angular velocity of the pedal sprocket.

In this exercise, the angular speed of the pedal sprocket is calculated as:

• \( \omega_{pedal} = 40 \times \frac{2\pi}{60} = 4.19 \) rad/min

This angular speed then helps determine how fast the wheel sprocket revolves.

Linear Velocity

Linear velocity relates the angular speed to the actual speed of the bicycle's movement across a surface. It is essentially the speed at which a point on the edge of a wheel moves, and it plays a vital role in making sense of bicycle speed in terms of real distances.

For two connected gears like the pedal sprocket and the wheel sprocket, the linear velocity at the edge of each gear must be the same because the chain transfers motion without slipping. The formula connecting angular speed and linear velocity is:

• \( v = r \times \omega \)

where \( v \) is linear velocity, \( r \) is radius, and \( \omega \) is angular speed.

Therefore, in the exercise, the linear speed is first calculated for the pedal sprocket and then equated to that of the wheel sprocket to find its angular speed in rpm, thereby helping us maintain consistency between the two connected gears.

Radians Conversion

Radians conversion is a method to translate revolutions or degrees into a universally applicable unit for measuring angles, which is crucial for calculations dealing with angular speed and rotation. One full revolution equals \( 2\pi \) radians.

In the context of this problem, radians help transition between rpm (revolutions per minute) and angular speed calculated in a usable form for further formulas. For instance, converting the pedal rotation:

• \( 40 \times \frac{2\pi}{60} = 4.19 \) rad/min

This conversion allows for direct use in equations that involve multiplying by radius to find linear velocities and ultimately, the bicycle's speed.

Understanding radians is essential as it forms a bridge from angular measures to linear and distance-related measures, especially when discussing rotational dynamics in precalculus.

Bicycle Speed Calculation

Bicycle speed calculation translates technical values like angular speed and linear velocity into comprehensible real-world speeds, such as miles per hour (mph). This transformation involves several steps: determining the angular speed of the wheel and using it to find how fast the bicycle travels.

First, assign the wheel's angular speed to the wheel sprocket's angular speed because they turn together. With the angular speed of the wheel sprocket known (80 rpm), convert this to the linear speed of the wheel:

• \( v = 13 \times \left( \frac{80 \times 2\pi}{60} \right) \approx 1,088 \) inches/min

Convert this linear speed from inches to feet, then from feet per minute to miles per hour:

• \( v_{feet/min} = \frac{1,088}{12} = 90.67 \) feet/min
• \( v_{mph} = \frac{90.67 \times 60}{5,280} \approx 1.03 \) mph

The bicycle thus moves at about 1.03 mph, which illustrates how rotational dynamics translate into practical movement speeds—a unique aspect of precalculus linked to real-world physics.