A long thin rod lies along the x-axis from the origin to x=L, with L= 0.890 m. The mass per unit length, λ (in kg/m) varies according to the equation λ = λ0 (1+1.410x2). The value of λ0 is 0.700 kg/m and x is in meters.
1. Calculate the total mass of the rod.
2. Calculate the x-coordinate of the center of mass of the rod.
3. Calculate the moment of inertia of the rod with respect to the y-axis.
Akko
To calculate the total mass of the rod, we need to integrate the mass per unit length, λ, over the length of the rod.
1. Total Mass of the Rod:
To calculate the total mass, we integrate the mass per unit length, λ, over the length of the rod:
M = ∫ λ dx where x varies from 0 to L.
Given λ = λ0 (1 + 1.410x^2), we substitute this into the equation:
M = ∫ (λ0 (1 + 1.410x^2)) dx
To solve the integral, we integrate each term separately:
M = λ0 ∫ (1 + 1.410x^2) dx
= λ0 (x + 1.410 * (x^3)/3) + C
Evaluating the integral with the limits of integration:
M = λ0[(L + 1.410 * (L^3)/3)] - λ0[(0 + 1.410 * (0^3)/3)]
= λ0[L + (1.410/3)L^3]
Substituting the given values:
λ0 = 0.700 kg/m
L = 0.890 m
M = 0.700[0.890 + (1.410/3)(0.890^3)]
= 0.700(0.890 + 0.702797)
= 0.700(1.592797)
= 1.1159589 kg
Therefore, the total mass of the rod is approximately 1.116 kg.
2. x-coordinate of the Center of Mass:
To calculate the x-coordinate of the center of mass, we use the formula:
x_com = ∫ x * λ dx / M, where x_com represents the x-coordinate of the center of mass.
We substitute the given values into the formula:
x_com = ∫ x * λ dx / M
= ∫ (x * λ0 (1 + 1.410x^2)) dx / 1.116
Again, we integrate each term separately:
x_com = λ0 ∫ (x + 1.410x^3) dx / 1.116
= λ0[(x^2/2) + (1.410/4)x^4] / 1.116
Evaluating the integral with the limits of integration:
x_com = λ0[((L^2/2) + (1.410/4)L^4) - ((0^2/2) + (1.410/4)(0^4))] / 1.116
= λ0[(L^2/2) + (1.410/4)L^4] / 1.116
Substituting the given values:
λ0 = 0.700 kg/m
L = 0.890 m
x_com = (0.700[(0.890^2/2) + (1.410/4)(0.890^4)]) / 1.116
Calculating the value gives us:
x_com ≈ 0.304 m
Therefore, the x-coordinate of the center of mass is approximately 0.304 meters.
3. Moment of Inertia with respect to the Y-axis:
The moment of inertia of the rod with respect to the y-axis can be calculated using the formula:
I_y = ∫ λ x^2 dx, where I_y represents the moment of inertia.
Substituting the given mass per unit length:
I_y = ∫ λ x^2 dx
= ∫ (λ0 (1 + 1.410x^2)) x^2 dx
Again, we integrate each term separately:
I_y = λ0 ∫ (1 + 1.410x^2) x^2 dx
= λ0[(x^3/3) + (1.410/5)x^5] + C
Evaluating the integral with the limits of integration:
I_y = λ0[((L^3/3) + (1.410/5)L^5) - ((0^3/3) + (1.410/5)(0^5))]
Substituting the given values:
λ0 = 0.700 kg/m
L = 0.890 m
I_y = 0.700[((0.890^3/3) + (1.410/5)(0.890^5))] - 0.700[((0^3/3) + (1.410/5)(0^5))]
Calculating the value gives us:
I_y ≈ 0.154 kg·m^2
Therefore, the moment of inertia of the rod with respect to the y-axis is approximately 0.154 kg·m^2.