Q46P Find the mass of a cube of side ... [FREE SOLUTION]

Chapter 5: Q46P (page 248)

Find the mass of a cube of side 2 if the density is proportional to the square of the distance from the center of the cube.

Short Answer

Expert verified

The required Mass of a cube is $8K$ .

Step by step solution

Definition of Mass

A physical body's mass is the amount of matter it contains. It's also a measure of inertia, or the resistance to acceleration when a net force is applied.

Finding the Mass of a cube

The origin should be in the center of the cube. We may compute the mass in one octant and then double the result by 8 because the cube's density is symmetrical in x, y and z .

Now, Find the Mass,

$M = 8 K {鈭�}_{0}^{1} (x^{2} + y^{2} + z^{2}) d x {鈭�}_{0}^{1} d y {鈭�}_{0}^{1} d z = {8 K {鈭�}_{0}^{1} d x {鈭�}_{0}^{1} d y (x^{2} z + y^{2} z + \frac{1}{3} z^{3})|}_{0}^{1} = 8 K {鈭�}_{0}^{1} d x {鈭�}_{0}^{1} d y (x^{2} + y^{2} + \frac{1}{3}) = {8 K {鈭�}_{0}^{1} d x (x^{2} y + \frac{1}{3} y^{3} + \frac{1}{3} y)|}_{0}^{1} = 8 K {鈭�}_{0}^{1} d x (x^{2} + \frac{1}{3} + \frac{1}{3})$

The final value is obtained using further calculation.

$M = 8 K {鈭�}_{0}^{1} (x^{2} + \frac{2}{3}) d x = {8 K [\frac{x^{3}}{3} + \frac{2}{3} x]|}_{0}^{1} = 8 K (\frac{1}{3} + \frac{2}{3}) = 8 K$

M = 8 K {鈭�}_{0}^{1} (x^{2} + \frac{2}{3}) d x = {8 K [\frac{x^{3}}{3} + \frac{2}{3} x]|}_{0}^{1} = 8 K (\frac{1}{3} + \frac{2}{3}) = 8 K

Therefore, the Mass of a cube is $8 K$ .

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91影视

Find the mass of a cube of side 2 if the density is proportional to the square of the distance from the center of the cube.

Short Answer

Step by step solution

Definition of Mass

Finding the Mass of a cube

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