91Ó°ÊÓ

Understanding how to calculate the volume of a cylinder is essential for this exercise. A cylinder's volume is determined by the formula: \( V = \pi r^2 h \). Let's break down what each part of this formula represents:

• \(\pi\) (Pi) is a constant approximately equal to 3.14159. It's crucial in calculations involving circles.
• \(r\) is the radius of the base of the cylinder. In this problem, the radius is 0.50 meters.
• \(h\) is the height of the cylinder. Here, it's given as 2.2 meters.

To find the volume, simply plug these values into the formula. First, square the radius (0.50 m): \(0.50^2 = 0.25\). Then multiply this by \(\pi\) and the height (2.2 m):
\[V = \pi \times 0.25 \times 2.2 \approx 1.73 \text{ m}^3\]
This result tells us the amount of space the pillar occupies. Calculating volume is an initial step in many physics problems, preparing us to find mass next.

To solve physics problems involving mass and weight, understanding the conversion between the two is vital. Mass is a measure of how much matter an object contains, usually given in kilograms. To find the mass from the volume, multiply the volume by the material's density:

• Density of concrete: \(2.2 \times 10^{3} \text{ kg/m}^3\)
• Volume of the pillar: \(1.73 \text{ m}^3\)

Multiply these values to get the mass:
\[\text{mass} = 2.2 \times 10^{3} \times 1.73 \approx 3806 \text{ kg}\]
At this point, you have the mass and need to convert it to weight, which is the force exerted by gravity on that mass. Weight (in Newtons) is calculated by multiplying mass by the gravitational acceleration \(g\), approximately \(9.81 \text{ m/s}^2\).
\[\text{weight (N)} = 3806 \times 9.81 \approx 37335 \text{ N}\]
Finally, convert the weight from newtons to pounds using the given conversion factor: 1 Newton = 0.2248 pounds.
\[\text{weight (lb)} = 37335 \times 0.2248 \approx 8393 \text{ lb}\]
This conversion step is crucial in translating metrics into units used in everyday contexts.

Density is a key concept when working out weight from volume. It represents how much mass is contained in a unit volume of a substance, typically expressed in kilograms per cubic meter (kg/m\(^3\)). It's important because it allows us to transition from volume—which was calculated from geometric properties—into mass.

• Higher density means more mass in a given volume. Thus, a dense material like concrete will have significant weight because it concentrates a lot of mass into its structure.
• The density of concrete, \(2.2 \times 10^{3} \text{ kg/m}^3\), indicates that every cubic meter of concrete weighs 2200 kg.

By understanding this, one can see that finding the weight isn't merely about size but also about what's contained within that size. Once density gives you the mass, weight comes from incorporating gravity—simply a force acting on that mass, pulling it toward the earth. These principles underpin much of physics when working with real-world applications, helping to bridge theoretical models and practical weight measurements.