To determine the moment of inertia of a rod with a non-uniform density that varies linearly from the hinged end to the free end, we need to consider how the mass distribution affects the overall moment of inertia. In this case, the density of the rod doubles from the hinge to the free end, which means we have to integrate to find the moment of inertia based on the changing density.
Understanding the Problem
We have a rod of length L and mass M, hinged at one end. The density, ρ, varies linearly from the hinge to the free end. If we denote the density at the hinge as ρ₀, then at the free end (length L), the density becomes 2ρ₀. This linear variation can be expressed mathematically.
Density Function
The density can be expressed as:
- ρ(x) = ρ₀ + (2ρ₀ - ρ₀) * (x/L) = ρ₀(1 + x/L)
Here, x is the distance from the hinge, ranging from 0 to L.
Calculating the Mass Element
To find the moment of inertia, we first need to calculate the mass element, dm, of a small segment of the rod at a distance x from the hinge. The length of this small segment, dx, gives us:
- dm = ρ(x) * dx = ρ₀(1 + x/L) * dx
Moment of Inertia Element
The moment of inertia dI of this small segment about the hinge is given by:
- dI = x² * dm = x² * ρ₀(1 + x/L) * dx
Integrating to Find Total Moment of Inertia
Now, we need to integrate dI from 0 to L to find the total moment of inertia I:
- I = ∫(from 0 to L) x² * ρ₀(1 + x/L) dx
This integral can be split into two parts:
- I = ρ₀ ∫(from 0 to L) x² dx + ρ₀/L ∫(from 0 to L) x³ dx
Evaluating the Integrals
Calculating the first integral:
- ∫(from 0 to L) x² dx = [x³/3] (from 0 to L) = L³/3
Now for the second integral:
- ∫(from 0 to L) x³ dx = [x⁴/4] (from 0 to L) = L⁴/4
Putting It All Together
Substituting these results back into our expression for I:
- I = ρ₀ (L³/3) + ρ₀/L (L⁴/4)
- I = ρ₀ (L³/3) + ρ₀ (L³/4)
Now, we can combine these terms:
- I = ρ₀ L³ (1/3 + 1/4) = ρ₀ L³ (4/12 + 3/12) = ρ₀ L³ (7/12)
Finding ρ₀ in Terms of M
To express the moment of inertia in terms of the total mass M, we need to find ρ₀. The total mass M can be calculated by integrating the density over the length of the rod:
- M = ∫(from 0 to L) ρ(x) dx = ∫(from 0 to L) ρ₀(1 + x/L) dx = ρ₀ (L + L/2) = 3ρ₀L/2
From this, we can solve for ρ₀:
Final Expression for Moment of Inertia
Substituting ρ₀ back into the moment of inertia equation:
- I = (2M/(3L)) * (7/12) * L³ = (7ML²)/18
Thus, the moment of inertia of the rod about the hinge point is:
This result shows how the non-uniform density affects the distribution of mass and, consequently, the moment of inertia. The integration process allows us to account for the varying density along the length of the rod, leading to a precise calculation of its rotational inertia.
