Question

A long thin solid rod lies along the positive x-axis. One end is at x = 1.80 m and the other at x = 3.40 m. The linear mass density is λ = a$x^{3}$ + bx, where λ is measured in kg/m, and the constants have the following values: a = 1.70 kg/$m^{4}$ and b = 2.20 kg/$m^{2}$.
1. Determine the total mass of the rod.
2. Calculate the x-coordinate of the center of the mass for this rod.

Answer 1

1. The total mass of the rod is 61.48 kg.

2. The x-coordinate of the center of the mass for this rod is  172.6021 m.

What is density?

The density is the ratio of the mass and the volume of the object. It is denoted by ρ.

ρ = mass/Volume = m/V

Given a long thin solid rod lies along the positive x-axis. One end is at x = 1.80 m and the other at x = 3.40 m. The linear mass density is

λ = ax³ + bx,

where λ is measured in kg/m, and the constants have the following values: a = 1.70 kg/m⁴ and b = 2.20 kg/m²

So, λ = 1.70x³ + 2.20x

1. The total mass of the rod is

m =₁.₈³⁴ ∫ [1.70x³ + 2.20x] dx

m = [ 1.70x⁴/4 + 2.20x² /2 ]₁.₈³⁴

Putting the limits, we have

m = 69.51 - 8.02548

m = 61.48 kg

Thus, the total mass of the rod is 61.48 kg

2. The x-coordinate of the center of the mass for this rod

Xcm =∫ xλdx/m

Xcm =∫[ x ( 1.70x³ + 2.20x) /m ]dx

Xcm =∫[( 1.70x⁴ + 2.20x²) /m ]dx

On integrating and putting the limits  x = 1.80 m to x = 3.40 m

Xcm = = [ {1.70(3.4)⁵/5 + 2.20(3.4)³ /3} - {1.70(1.8)⁵/5 + 2.20(1.8)³ /3} ]

Xcm = 183.3034 - 10.70133

Xcm = 172.6021 m

Thus,  the x-coordinate of the center of the mass for this rod is  172.6021 m.

Learn more about density.

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