small cube of copper (density = 8930 kg/m3) is used to support a large cube of marble (density = 2560 kg/m3). The copper cube has sides of length 2.00 cm and is carefully placed on a flat, level table under the center of the cube of marble that has sides of length 30.0 cm. Given that the Young's modulus of copper is 110 GPa, by how much is the copper compressed? Answer is 3.1 x10-8 m. How?
To find by how much the copper cube is compressed, you can use Hooke's Law, which states that the strain (change in length) of a material is directly proportional to the stress (force applied) and inversely proportional to the Young's modulus of the material.
Let's break down the steps to find the answer:
1. Determine the volume of the copper cube:
The volume of a cube is calculated by cubing the length of its side. In this case, the copper cube has sides of length 2.00 cm, which translates to a volume of (0.02 m)^3 = 8.00 x 10^-6 m^3.
2. Determine the mass of the copper cube:
The density of copper is given as 8930 kg/m^3. The density of an object can be calculated by dividing its mass by its volume. Rearranging the formula, we can determine the mass of the copper cube: Mass = Density x Volume. Thus, the mass of the copper cube is 8930 kg/m^3 * 8.00 x 10^-6 m^3 = 0.071 kg.
3. Determine the force exerted by the marble cube on the copper cube:
The force exerted by an object can be calculated by multiplying its mass by the acceleration due to gravity. In this case, the force exerted on the copper cube is 0.071 kg * 9.8 m/s^2 = 0.6978 N (rounded to four decimal places).
4. Determine the area of the copper cube:
Since the copper cube is supporting the marble cube from underneath, the area of contact between the two cubes is equal to the cross-sectional area of the copper cube. The cross-sectional area of a cube is calculated by squaring the length of the side. Thus, the area of the copper cube is (0.02 m)^2 = 4.0 x 10^-4 m^2.
5. Determine the stress applied to the copper cube:
Stress is defined as the force applied per unit area. In this case, the stress applied to the copper cube is calculated by dividing the force (0.6978 N) by the area (4.0 x 10^-4 m^2). Thus, the stress on the copper cube is 0.6978 N / 4.0 x 10^-4 m^2 = 1745 N/m^2.
6. Determine the compression (strain) of the copper cube:
Finally, using Hooke's Law and the known Young's modulus of copper (110 GPa = 110 x 10^9 N/m^2), we can calculate the strain (change in length) of the copper cube. The strain is equal to the stress divided by the Young's modulus:
Strain = Stress / Young's modulus = 1745 N/m^2 / (110 x 10^9 N/m^2) = 1.587 x 10^-11.
7. Convert the strain to a change in length:
The strain is defined as the change in length divided by the original length. Since the length of the copper cube didn't change in the other dimensions, the change in length is equal to the strain multiplied by the original length:
Change in length = Strain * Original length = 1.587 x 10^-11 * 0.02 m = 3.174 x 10^-13 m.
Therefore, the copper cube is compressed by 3.174 x 10^-13 m, which can be expressed as 3.1 x 10^-8 m in scientific notation.