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Chapter 1: Problem 35

A spacecraft component occupies a volume of \(8 \mathrm{ft}^{3}\) and weighs \(25 \mathrm{lb}\) at a location where the acceleration of gravity is \(31.0 \mathrm{ft} / \mathrm{s}^{2}\). Determine its weight, in pounds, and its average density, in \(\mathrm{lbm} / \mathrm{ft}^{3}\), on the moon, where \(g=5.57 \mathrm{ft} / \mathrm{s}^{2}\).

Short Answer

Expert verified
The weight of the object on the moon is 4.49 lb and its average density there is 3.25 lbm/ft^{3}

Step by step solution

01

Compute the mass

The weight of the object can be found by dividing the weight by the acceleration due to gravity. This gives the mass \(m\) in slugs. Using the first known location: \( m = F / g = 25 lb / 31.0 ft/s^{2} = 0.80645 slugs \)
02

Calculate the Weight on the Moon

The weight on the moon can be calculated using the mass found in step 1 and the known gravity on the moon. \( F = m * g = 0.80645 slugs * 5.57 ft/s^{2} = 4.49 lb \)
03

Convert the Mass to Pounds mass

You can convert mass in slugs to mass in pounds mass (lbm) knowing that 1 slug = 32.2 lbm. So, \(m = 0.80645 slugs * 32.2 lbm/slug = 25.97 lbm \)
04

Calculate the Average Density

The average density can be found by dividing the mass by the volume of the object. \( rho = m / V = 25.97 lbm / 8.0 ft^{3} = 3.25 lbm/ft^{3} \)

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Key Concepts

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

Mass and Weight Conversion
Understanding the distinction between mass and weight is crucial in physics and engineering. Mass is a measure of the amount of matter in an object and does not change regardless of location. Weight, however, is the force exerted by gravity on that mass and can change with location. To find the mass when the weight is given, divide the weight by the acceleration due to gravity at that specific location. This conversion was used in our exercise to calculate the spacecraft component's mass.

For instance, a spacecraft component that weighs 25 pounds on Earth can be converted to mass in slugs by dividing by the acceleration due to gravity (usually 32.174 ft/s虏 on Earth), not the 31.0 ft/s虏 used in our particular example. Once in slugs, the mass can be further converted to pounds mass (lbm) using the relation that 1 slug equals 32.2 lbm. This conversion is particularly useful when dealing with different celestial bodies, like the Moon, where gravity's acceleration is not the same as on Earth.
Acceleration Due to Gravity
Acceleration due to gravity, often denoted as 'g', varies depending on location. On Earth, 'g' is approximately 32.174 ft/s虏, but it is different on the Moon, other planets, or in outer space.

When calculating the weight of an object on the Moon, as shown in our exercise, you need to use the Moon's 'g' which is 5.57 ft/s虏. Weight, being a force, is the product of mass and acceleration due to gravity (F = m*g). Therefore, knowing the mass, we can compute the weight on the Moon using the Moon's 'g'. This calculation allows scientists and engineers to anticipate how objects will behave in different gravitational environments, essential for space exploration.
Average Density Computation
Average density is the measure of how much mass there is in a given volume and is crucial in fields ranging from material science to geology and astronomy. To calculate the average density, divide mass (in lbm in the context of this exercise) by volume (in ft鲁).

In our exercise, the average density serves to understand the compactness of a spacecraft component in different environments. Knowing the density helps determine if an object will float or sink in a fluid and can influence the design of structures or vehicles that need to perform under varying atmospheric conditions or within fluids of different densities.

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Most popular questions from this chapter

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