the moment of inertia of a thin uniform rod of mass M and length L about an axis passing through a point midway between the centre and one end perpendicular to its length is -----------------------------------
To find the moment of inertia of a thin uniform rod about an axis passing through a point midway between the center and one end perpendicular to its length, you can follow these steps:
1. Identify the given variables:
- Mass of the rod, M
- Length of the rod, L
2. Determine the formula for the moment of inertia of a thin rod about an axis perpendicular to its length and passing through one end. This formula is given by:
I = (1/3) * M * L^2
3. To find the moment of inertia about an axis passing through a point midway between the center and one end (let's call it axis A), we can use the parallel axis theorem. According to the theorem, the moment of inertia about a parallel axis is equal to the moment of inertia about a centroidal axis (a axis passing through the center of mass) plus the product of the mass and the square of the perpendicular distance between the axes.
4. Let's denote the moment of inertia about the centroidal axis (axis C) as IC, and the perpendicular distance between axes C and A as d.
5. Since the rod is thin and uniform, the center of mass coincides with the centroid. Therefore, the distance d will be half of the length of the rod, L/2.
6. Applying the parallel axis theorem, the moment of inertia about axis A is given by:
IA = IC + M * d^2
7. Substituting the values, we have:
IA = IC + M * (L/2)^2
8. From step 2, we know that the moment of inertia about the centroidal axis (IC) is (1/3) * M * L^2. Substituting this into the equation, we can simplify:
IA = (1/3) * M * L^2 + M * (L/2)^2
= (1/3) * M * L^2 + (1/4) * M * L^2
9. Combining the like terms, we get:
IA = (7/12) * M * L^2
Therefore, the moment of inertia of a thin uniform rod of mass M and length L about an axis passing through a point midway between the center and one end, perpendicular to its length, is (7/12) * M * L^2.