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A designer has proposed to use a propellant grain with a rectangular cross-section (0.5 m by 0.3 m), an outer diameter of 1.1 m, and a length of 1.5 m. If the density of the propellant is 1450 kg/m\(^3\), determine the sliver mass of the propellant grain and the percentage of the sliver mass compared to the initial propellant mass at the end of its operation. By restricting the fraction of sliver mass to 5%, can it be used for practical application?

Answer :

Final answer:

First, calculate the volume of the propellant grain via the formula for the volume of a cylinder, then determine the mass of the propellant by multiplying the volume by the given density.

Explanation:

To begin with, let's calculate the mass of the propellant. The volume of the propellant grain can be calculated using the formula for the volume of a cylinder (V = πr²h) where r is the radius and h is the height. The radius of the propellant grain is given as 0.55 meters and the height as 1.5 meters. Hence, the volume of the propellant grain is V = π(0.55 m)²(1.5 m).

The mass of the propellant can be calculated by multiplying its volume by its density (m = ρV), where ρ represents the density of the propellant, which is given as 1450 kg/m³. So, the mass of the propellant is m = 1450 kg/m³ * V. According to the given data, 98.9% of the rocket is fuel, or we can say that 1.10% is sliver mass.

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