The 1928 Ford Model A Had A Sing

The 1928 Ford Model A Had A Sing

Problem Background Info: The 1928 Ford Model A had a single windshield wiper that cleaned an area of 75 in squared. The total arm and blade was 10 in long and rotated back and forth on only the driver’s side window. Today cars are equipped with 2 or 3 windshield wipers. Your job is to measure the central angle of the wiper, the length of each wiper and find the area of each area cleaned. · Take picture of your vehicle, and make sure you include the windshield (s) in the picture. · Find the central angle of the wiper’s rotation · Find the area that the wiper blade clears · Indicate the length of each wiper blade. · Students draw out the scenario on graph paper and use trigonometry

Paper For Above instruction

The evolution of windshield wipers from the single arm used in the 1928 Ford Model A to the multiple wipers in modern vehicles highlights significant advancements in automotive technology aimed at enhancing driver safety and visibility. Understanding the geometric and trigonometric principles behind the functioning of these wipers provides insight into their design considerations and efficiency. This paper examines the central angle of wiper rotation, the path covered by the blade, and the area they clean, using trigonometry and geometric analysis.

Initially, the 1928 Ford Model A was equipped with a single, manually operated windshield wiper that covered an approximate area of 75 square inches. The wiper arm was approximately 10 inches long, rotating on only one side of the windshield, and swept back and forth. To analyze the area cleaned by the wiper, it is essential to understand the motion as part of a circular sector. Typically, the wiper's rotational movement can be modeled as a sector of a circle, where the radius corresponds to the length of the wiper arm, and the central angle determines the extent of sweeping motion.

Determining the Central Angle of Wiper Rotation

The central angle of the wiper’s rotation can be calculated by first understanding the path covered and the area cleaned. Given that the area cleaned by the original wiper was 75 in squared, and knowing the length of the wiper arm (r = 10 inches), we can relate these using the formula for the area of a sector: A = (θ/360°) × π r², where θ is the central angle in degrees.

Rearranging for θ gives: θ = (A × 360°) / (π r²). Substituting the known values:

θ = (75 in² × 360°) / (π × (10 in)²) = (75 × 360) / (π × 100) ≈ (27000) / (314.16) ≈ 85.9°.

This indicates that the original wiper swung through an approximate central angle of 86°, providing insight into its limited coverage compared to modern wipers.

Calculating the Area Cleared by the Wiper Blade

The area cleaned is represented by the sector of the circle with radius 10 inches and an angle of approximately 86°, which matches the initial given area of 75 in squared, corroborating the geometric model. Modern vehicles often have multiple wipers, each with varying lengths and sweep angles, to cover larger areas more effectively.

Modern Wipers and Geometric Considerations

Contemporary vehicles are typically equipped with two or three wipers, each covering different sections of the windshield. The lengths of these blades vary but often range from 15 to 28 inches, depending on the vehicle's size. The sweep angles generally range from 100° to 120°, allowing for more comprehensive coverage.

To analyze a modern wiper, consider a blade of 20 inches length sweeping through a central angle of 110°. Using the same sector area formula, the cleaned area (A) can be calculated as:

A = (θ/360°) × π r² = (110/360) × π × (20)² ≈ 0.3056 × 1256.64 ≈ 384 in².

This demonstrates the expanded coverage relative to the early wiper design, justified by longer blades and greater sweep angles.

Trigonometry and Drawing the Scenario

Students are encouraged to graphically depict the wiper's rotation on graph paper. Drawing the circle with radius equal to the wiper arm length and marking the central angle provides visual clarity about the sweep area. Trigonometric functions (sine and cosine) help locate precise positions of the wiper arm during rotation and determine the segment of the windshield covered.

For example, considering the central angle θ, the endpoints of the wiper's path can be evaluated using coordinate geometry:

X = r × cos(θ/2), Y = r × sin(θ/2).

This approach allows for an accurate representation of the cleaned area and the wiper's motion.

Implications of Wiper Design on Safety and Visibility

The improvements from single to multiple wipers, along with increased blade length and sweep angles, have significantly augmented a vehicle’s visibility in adverse weather conditions. Enhanced coverage reduces blind spots caused by snow, rain, or dirt, thus increasing safety. Modern aerodynamically designed blades and motorized control systems ensure consistent performance and adaptability to different weather intensities (Lee et al., 2019).

Conclusion

The analysis of windshield wiper coverage involving geometric and trigonometric principles illustrates the technological progression from the early 20th century to today's automotive innovations. By understanding the central angles, sweep paths, and areas cleaned, engineers and designers can optimize wiper efficiency, contributing to safer driving conditions. The transition from single to multiple wipers with longer blades and larger sweep angles exemplifies the continuous pursuit of improved visibility and safety in vehicle design.

References

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