91Ó°ÊÓ

To find the volume of a cylinder, which is the shape of the concrete pillar, you need to apply the mathematical formula for the volume of a cylinder: \( V = \pi r^2 h \). Here, \( V \) represents the volume, \( r \) stands for the radius, and \( h \) is the height of the cylinder. The inputs here are the radius \( r = 0.50 \, \text{m} \) and the height \( h = 2.2 \, \text{m} \). This particular formula is derived from the fact that the area of the circular base is \( \pi r^2 \) and needs to be multiplied by the height to account for the whole cylinder.

• Substitute the given values into the formula: \( V = \pi \times (0.50)^2 \times 2.2 \).
• After simplifying, you find that \( V = 3.14 \times 0.25 \times 2.2 = 1.727 \, \text{m}^3 \).

This step of finding the volume is crucial because it forms the basis for determining the pillar's mass, using density, in the next step.

Density is a key physical property that connects the mass of an object with its volume. The formula for density \( \rho \) is \( \rho = \frac{m}{V} \), which states that density is the mass \( m \) divided by the volume \( V \). In this problem, we work in the reverse direction to find mass from given density and calculated volume: \( m = \rho \times V \).

• The known density of concrete is \( 2.2 \times 10^3 \, \text{kg/m}^3 \).
• You previously calculated the volume as \( 1.727 \, \text{m}^3 \).
• Thus, the mass is calculated by multiplying density by volume: \[ m = 2.2 \times 10^3 \times 1.727 = 3799.4 \, \text{kg} \].

This step showcases the importance of understanding material properties and how they influence overall weight calculations of objects in physics.

Weight is the force due to gravity acting on an object's mass. Using the earth's gravitational acceleration \( g \approx 9.81 \, \text{m/s}^2 \), the weight \( W \) is given by \( W = m \times g \).

• Given the mass from the previous steps, \( m = 3799.4 \, \text{kg} \).
• Calculate the weight in newtons: \[ W = 3799.4 \times 9.81 = 37243.014 \, \text{N} \].

Since the problem requires the answer in pounds, we use conversion: \( 1 \, \text{N} = 0.2248 \, \text{lb} \).

• Convert the weight: \[ W = 37243.014 \times 0.2248 = 8368.06 \, \text{lb} \].

Weight conversion is a powerful tool in physics that allows us to express computations in different units of measurement to fit practical needs or regional standards.

In physics, problem-solving requires understanding how to apply mathematical formulas to real-world situations. Simplifying complex issues into step-by-step solutions is essential.

• Start by understanding the problem and identifying known variables, such as the dimensions of the cylinder and the density of the material.
• Use basic principles, like calculating volume and applying density formulas, to find intermediate values like mass.
• Translate these values into what is required, such as converting mass into weight using gravitational acceleration.
• Finally, convert units if necessary to satisfy the problem's output requirements, such as converting newtons to pounds.

Being able to break down complex situations into simple steps not only aids in solving the problems at hand but also helps in developing a deeper understanding of physical concepts and their applications.