A cube 5.0 cm on each side is made of a metal alloy. After you drill a cylindrical hole 2.0 cm in diameter all the way through and perpendicular to one face, you find that the cube weighs 6.30 N. What is the density of this metal?

검증된 단계별 안내

Calculate the volume of the cube before the hole is drilled. The formula for the volume of a cube is \( V_{cube} = s^3 \), where \( s \) is the side length. Here, \( s = 5.0 \) cm, so \( V_{cube} = (5.0 \text{ cm})^3 \).

Calculate the volume of the cylindrical hole. The formula for the volume of a cylinder is \( V_{cylinder} = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height. The diameter of the cylinder is 2.0 cm, so the radius \( r = 1.0 \) cm, and the height \( h \) is the same as the side of the cube, 5.0 cm.

Subtract the volume of the cylindrical hole from the volume of the cube to find the volume of the remaining metal. \( V_{metal} = V_{cube} - V_{cylinder} \).

Use the weight of the cube to find the mass. The weight \( W \) is given as 6.30 N. Use the relation \( W = mg \) to find the mass \( m \), where \( g \) is the acceleration due to gravity (approximately 9.81 m/s²). Rearrange to find \( m = \frac{W}{g} \).

Calculate the density of the metal using the formula \( \rho = \frac{m}{V_{metal}} \), where \( \rho \) is the density, \( m \) is the mass, and \( V_{metal} \) is the volume of the remaining metal.

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주요 개념

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Density

Density is a measure of mass per unit volume, typically expressed in kilograms per cubic meter (kg/m³). It is calculated by dividing the mass of an object by its volume. Understanding density is crucial for determining how much mass is contained in a given volume of material, which is essential for solving problems involving mass and volume relationships.

Volume of a Cube and Cylinder

The volume of a cube is calculated by cubing the length of one of its sides (V = a³), where 'a' is the side length. For a cylinder, the volume is found using the formula V = πr²h, where 'r' is the radius and 'h' is the height. In this problem, understanding these formulas helps determine the total volume of the cube and the volume of the cylindrical hole drilled through it.

Buoyancy and Weight

Weight is the force exerted by gravity on an object, calculated as the product of mass and gravitational acceleration (W = mg). In this context, the weight of the cube after drilling the hole is given, which allows for the calculation of the mass. Understanding the relationship between weight, mass, and volume is essential for determining the density of the metal alloy.