Problem 253 - Precalculus

Question

The height of a piston, \(h,\) in inches, can be modeled by the equation \(y=2 \cos x+6,\) where \(x\) represents the crank angle. Find the height of the piston when the crank angle is \(55^{\circ} .\)

Step-by-Step Solution

Answer

The height of the piston is approximately 7.15 inches.

Step 1: Substitute the Crank Angle into the Model

Begin by substituting the given crank angle into the equation: \( y = 2\cos x + 6 \). Since the crank angle \( x = 55^{\circ} \), we replace \( x \) with \( 55^{\circ} \) in the equation. This gives us \( y = 2\cos 55^{\circ} + 6 \).

Step 2: Evaluate the Cosine Function

Now, compute \( \cos 55^{\circ} \). Using a calculator, you find that \( \cos 55^{\circ} \approx 0.5736 \). Substitute this value back into the equation to get \( y = 2(0.5736) + 6 \).

Step 3: Calculate the Piston Height

Multiply \( 2 \) by \( 0.5736 \), which equals \( 1.1472 \). Add \( 1.1472 \) to \( 6 \) to find the height of the piston: \( y = 1.1472 + 6 \approx 7.1472 \). Thus, the height of the piston when the crank angle is \( 55^{\circ} \) is approximately \( 7.15 \) inches.

Key Concepts

Cosine Function

The cosine function is one of the basic trigonometric functions. It relates the angle and the lengths of the adjacent side to the hypotenuse in a right triangle. This function is particularly important when dealing with angles and circles, which frequently appear in physics and engineering problems. For any angle \( x \), the cosine function can be defined using the unit circle, where \( \cos x \) represents the x-coordinate of a point traveling counter-clockwise around the circle. In the piston and crankshaft mechanism, the cosine function models the repetitive up and down motion. This pattern of motion is periodic, making functions like cosine ideal due to their wave-like characteristics. The angle for cosine here is measured in degrees or radians, and the function takes an angle and gives a number ranging from \(-1\) to \(1\). This output helps in modeling phenomena such as piston motion precisely.

Piston Motion

Piston motion is a critical concept in understanding the operations of engines and compressors. The piston moves up and down within a cylinder as the engine's crankshaft rotates. This motion converts rotational energy into linear energy or vice versa, enabling the engine to perform work. In many engines, pistons are driven by crank mechanisms, causing them to slide smoothly up and down inside their cylinders. The crankshaft’s rotation creates a cyclical motion, making it highly relevant to model using trigonometric functions such as the cosine function. Understanding piston motion is essential because it directly impacts the efficiency and performance of engines. Proper modeling with equations can predict behaviors, potential stresses, and overall efficiency of the piston in the mechanical setup.

Crank Angle

The crank angle is an essential measure in the world of mechanical motion, especially for engines and pumps. It refers to the angle turned by the crank from a reference point, usually where the piston is at the top of its stroke. This angle determines the piston’s exact position in its cycle of motion. Being able to calculate or know the crank angle is vital in many applications:

• It helps in timing the spark in internal combustion engines.
• Understanding crank angle can also optimize the engine performance and reduce wear and tear.
• It aids in anticipating the precise movement of pistons through the cycle.

Knowing the crank angle allows for a more accurate model of piston behavior within an engine, providing insights into operational dynamics and potential areas for improvement.

Modeling with Equations

Modeling with equations is a powerful tool to describe and predict the behavior of physical systems. In the example of piston motion, equations like \( y = 2\cos x + 6 \) provide a mathematical representation of how the piston moves as the crankshaft turns. By substituting specific values into the equation, such as the crank angle, we can determine specific outcomes, like piston height. Advantages of using such models include:

• Ability to simulate different conditions without physical trial and error.
• Offers predictions of system behavior over time, helping in preventive maintenance.
• Enables visualization and understanding of complex systems.

This trigonometric modeling helps engineers and scientists design better machinery, test performances, and implement corrections efficiently. Understanding these equations and the ability to work with them is crucial in fields that rely heavily on rotational dynamics and periodic motions.