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Chapter 11: Problem 1

One of the concrete pillars that support a house is \(2.2 \mathrm{m}\) tall and has a radius of \(0.50 \mathrm{m}\). The density of concrete is about \(2.2 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3} .\) Find the weight of this pillar in pounds \((1 \mathrm{N}=0.2248 \mathrm{lb})\)

Short Answer

Expert verified
The weight of the pillar is approximately 8397 lbs.

Step by step solution

01

Calculate the Volume of the Cylinder

The pillar is shaped like a cylinder. The formula for the volume of a cylinder is \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height. Here, \( r = 0.50 \) m and \( h = 2.2 \) m. Plug these values into the formula: \[ V = \pi (0.50)^2 (2.2) \approx 1.732 \text{ m}^3 \]
02

Calculate the Mass of the Concrete

To find the mass, use the formula \( \text{mass} = \text{density} \times \text{volume} \). The density of concrete is given as \( 2.2 \times 10^3 \text{ kg/m}^3 \). Multiply this by the volume calculated:\[ \text{mass} = (2.2 \times 10^3) \times 1.732 \approx 3810.4 \text{ kg} \]
03

Calculate the Weight in Newtons

Weight is calculated using the formula \( \text{weight} = \text{mass} \times g \), where \( g \) is the acceleration due to gravity (\( \approx 9.81 \text{ m/s}^2 \)). Plug in the mass:\[ \text{weight} = 3810.4 \times 9.81 \approx 37371.5 \text{ N} \]
04

Convert the Weight to Pounds

To convert the weight from Newtons to pounds, use the conversion factor \( 1 \text{ N} = 0.2248 \text{ lb} \). Multiply the weight in Newtons by this factor:\[ \text{weight in lb} = 37371.5 \times 0.2248 \approx 8397 \text{ lb} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Volume of a Cylinder
The first step in solving our physics problem is to calculate the volume of the cylinder. A concrete pillar, like the one described, is shaped as a cylinder, and finding its volume involves a simple mathematical formula. The formula for the volume of a cylinder is given as:
  • \( V = \pi r^2 h \)
Here, \( r \) is the radius and \( h \) is the height of the cylinder. Plug in the known values: radius \( r = 0.50 \) meters and height \( h = 2.2 \) meters. Calculating the volume involves squaring the radius, multiplying by the height, and then by \( \pi \) (approximately 3.14159). Thus:
  • \( V = \pi (0.50)^2 (2.2) \approx 1.732 \text{ m}^3 \)
This calculation gives the space that the cylinder occupies in the three-dimensional world, measured in cubic meters.
Density and Mass Calculation
Knowing the volume, we next need to find the mass of the concrete pillar. Physics often involves finding the relationship between mass, volume, and density. The formula to connect these physical properties is:
  • \( \text{mass} = \text{density} \times \text{volume} \)
Here, the density is provided as \( 2.2 \times 10^3 \text{ kg/m}^3 \). By multiplying the density of the material by the previously calculated volume, we can find the mass:
  • \( \text{mass} = (2.2 \times 10^3) \times 1.732 \approx 3810.4 \text{ kg} \)
This means the concrete pillar has a mass of approximately 3810.4 kilograms. Density is a crucial physical property that defines how much mass exists in a specific volume.
Force and Weight Conversion
Once we have the mass, we need to find the weight of the object, and physics tells us that weight is a type of force. In scientific terms, weight is calculated by multiplying the mass by the acceleration due to gravity (usually \( g \approx 9.81 \text{ m/s}^2 \)):
  • \( \text{weight} = \text{mass} \times g \)
Substitute the mass of 3810.4 kg into the equation:
  • \( \text{weight} = 3810.4 \times 9.81 \approx 37371.5 \text{ N} \)
This calculation shows that the force exerted by the pillar, under Earth's gravity, is approximately 37371.5 Newtons. To make this number more comprehensible in everyday terms, we often need to convert it into pounds.
Unit Conversion in Physics
The final step in our problem is converting the weight from Newtons to pounds. Conversion between metric and imperial units is common in solving physics problems as different regions use different systems. The conversion factor we use here is:
  • \( 1 \text{ N} = 0.2248 \text{ lb} \)
By multiplying the weight in Newtons by this conversion factor, we translate it into a more familiar unit for some people:
  • \( \text{weight in lb} = 37371.5 \times 0.2248 \approx 8397 \text{ lb} \)
Thus, the weight of the concrete pillar is approximately 8397 pounds. Feel confident converting units: it ensures correct communication and uses the most relevant figures for your audience or problem context.

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Most popular questions from this chapter

In the process of changing a flat tire, a motorist uses a hydraulic jack. She begins by applying a force of \(45 \mathrm{N}\) to the input piston, which has a radius \(r_{1} .\) As a result, the output plunger, which has a radius \(r_{2},\) applies a force to the car. The ratio \(r_{2} / r_{1}\) has a value of \(8.3 .\) Ignore the height difference between the input piston and output plunger and determine the force that the output plunger applies to the car.

A hand-pumped water gun is held level at a height of \(0.75 \mathrm{m}\) above the ground and fired. The water stream from the gun hits the ground a horizontal distance of \(7.3 \mathrm{m}\) from the muzzle. Find the gauge pressure of the water gun's reservoir at the instant when the gun is fired. Assume that the speed of the water in the reservoir is zero and that the water flow is steady. Ignore both air resistance and the height difference between the reservoir and the muzzle.

A full can of black cherry soda has a mass of \(0.416 \mathrm{kg}\). It contains \(3.54 \times 10^{-4} \mathrm{m}^{3}\) of liquid. Assuming that the soda has the same density as water, find the volume of aluminum used to make the can.

The aorta carries blood away from the heart at a speed of about \(40 \mathrm{cm} / \mathrm{s}\) and has a radius of approximately \(1.1 \mathrm{cm} .\) The aorta branches eventually into a large number of tiny capillaries that distribute the blood to the various body organs. In a capillary, the blood speed is approximately \(0.07 \mathrm{cm} / \mathrm{s},\) and the radius is about \(6 \times 10^{-4} \mathrm{cm} .\) Treat the blood as an in compressible fluid, and use these data to determine the approximate number of capillaries in the human body.

An object is solid throughout. When the object is completely submerged in ethyl alcohol, its apparent weight is \(15.2 \mathrm{N}\). When completely submerged in water, its apparent weight is \(13.7 \mathrm{N}\). What is the volume of the object?
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