Problem 10 - Applied Physics

Question

Find the cross-sectional area of a piston head with a diameter of \(3.25 \mathrm{~cm}\).

Step-by-Step Solution

Answer

The cross-sectional area is approximately 8.30 cm².

Step 1: Understand the Problem

We need to find the cross-sectional area of a piston head. The cross-section of the piston head is circular, with a given diameter, which is 3.25 cm. To find the area, we will use the formula for the area of a circle.

Step 2: Recall the Formula for the Area of a Circle

The area \( A \) of a circle can be calculated using the formula \( A = \pi r^2 \), where \( r \) is the radius of the circle. The radius is half of the diameter.

Step 3: Calculate the Radius

The radius \( r \) of the circle is half of the diameter. Given the diameter \( d = 3.25 \) cm, the radius is \( r = \frac{3.25}{2} = 1.625 \) cm.

Step 4: Calculate the Area

With the radius \( r = 1.625 \) cm, substitute this value into the area formula \( A = \pi r^2 \). Thus, \( A = \pi (1.625)^2 \).

Step 5: Simplify the Calculation

Compute \( 1.625^2 = 2.640625 \), then multiply by \( \pi \) to get the area. So, \( A \approx 2.640625 \times 3.14159 \approx 8.30 \) cm².

Key Concepts

Circle Radius

To comprehend the notion of circle radius, let's begin with understanding the basics of a circle. The radius is a crucial aspect of a circle, which is the distance from the center point to any point on its boundary. It's half the length of the diameter, which is the line that goes through the center and touches two points on the edge of a circle.

So, if you're given the diameter, it's simple to find the radius. Just divide the diameter by two. For instance, if a circle has a diameter of 3.25 cm, the radius would be \( r = \frac{3.25}{2} = 1.625 \) cm. The radius is instrumental to many calculations in geometry, especially when determining the area, as it directly affects the value.

Understanding how to find the radius from the diameter helps solve various real-world problems, like calculating the size of different circular objects such as wheels, rings, or, like in our exercise, a piston head.

Piston Head

A piston head is a vital component in machines that utilize pistons, such as engines and pumps. It is typically circular and moves up and down within a cylinder, helping to convert energy into motion or vice versa.

The cross section of a piston head is of particular interest as it plays a significant role in the engine's performance. The cross-sectional area determines how much fluid or gas can exert force onto the piston head, impacting the movement and efficiency of the piston.

• A larger cross-sectional area allows more fluid pressure to be exerted, potentially leading to more forceful motion.
• Meanwhile, a smaller area might result in less motion.

Thus, understanding the dimensions of a piston head, such as its diameter and derived radius, is crucial when designing engines or assessing their performance.

Area of a Circle

The area of a circle is a critical concept in geometry, often used to determine how much space the circle occupies on a plane. Crucially, it is the space within the boundary of the circle itself.

The formula used to find the area is \( A = \pi r^2 \), where \( A \) represents the area, \( \pi \) is approximately 3.14159, and \( r \) is the radius of the circle. To determine the area accurately, first ensure you're correctly calculating the radius.

This formula shows the relationship between the radius and the area. A smaller radius will result in a smaller area, while a larger radius increases the area significantly since the radius is squared in the formula. In practical applications, understanding the area of circular objects, like piston heads, helps determine capacity and performance based on how much space is available within that circle.