Imagine a hollow sphere of radius R and mass M. It is welded to the end of a solid rod of length L, whose mass is also M. The rod is attached to the sphere in such a way that the whole assembly, when viewed from the side, looks like a lollipop. (That is, the axis of the rod, if extended, would pass through the center of the sphere.) Find the moment of inertia of the assembly about an axis perpendicular to the rod and passing through the rod's other end (the one away from the sphere).
Ah, the sweet lollipop assembly! Let's calculate its moment of inertia, shall we?
First, let's break it down into two separate parts: the hollow sphere and the solid rod.
For the hollow sphere, we know that the moment of inertia of a hollow sphere about an axis passing through its center is (2/3) * m * R^2, where m is the mass.
Now, for the solid rod, we can use the formula for the moment of inertia of a rod rotating about its end, which is (1/3) * m * L^2.
Since the mass of both the sphere and the rod is M, we can now calculate the total moment of inertia:
Total moment of inertia = Moment of inertia of hollow sphere + Moment of inertia of solid rod
Total moment of inertia = (2/3) * M * R^2 + (1/3) * M * L^2
And there you have it! The moment of inertia of the lollipop assembly about an axis perpendicular to the rod and passing through the rod's other end is (2/3) * M * R^2 + (1/3) * M * L^2.
Hope that brings a sweet smile to your face!
To find the moment of inertia of the assembly, we need to consider the contributions from both the hollow sphere and the solid rod.
1. Moment of Inertia of the Hollow Sphere:
The moment of inertia of a hollow sphere about an axis passing through its center is given by the formula:
I_sphere = (2/3) * M * R^2
2. Moment of Inertia of the Solid Rod:
The moment of inertia of a solid rod about an axis through one end and perpendicular to its length is given by the formula:
I_rod = (1/3) * M * L^2
3. Total Moment of Inertia of the Assembly:
Since the axis of rotation is perpendicular to the rod and passes through the other end of the rod, we can sum the moments of inertia of the sphere and the rod using the parallel axis theorem.
The parallel axis theorem states that if the distance between the axis of rotation and the center of mass is d, then the moment of inertia about the new axis is given by:
I_new_axis = I_center_of_mass + M * d^2
In this case, the distance between the axis of rotation and the center of mass of the rod is L, so:
I_rod_new = I_rod + M * L^2 = (1/3) * M * L^2 + M * L^2 = (4/3) * M * L^2
Now, the center of mass of the hollow sphere is at a distance L + R from the axis of rotation. So:
I_sphere_new = I_sphere + M * (L + R)^2 = (2/3) * M * R^2 + M * (L + R)^2
Finally, we can add the moments of inertia of the sphere and the rod to get the total moment of inertia of the assembly:
I_total = I_sphere_new + I_rod_new = (2/3) * M * R^2 + M * (L + R)^2 + (4/3) * M * L^2
Simplifying further, we have:
I_total = (2/3) * M * R^2 + M * (L + R)^2 + (4/3) * M * L^2
= (2/3) * M * R^2 + M * (L^2 + 2LR + R^2) + (4/3) * M * L^2
= (2/3) * M * R^2 + M * L^2 + 2M * LR + M * R^2 + (4/3) * M * L^2
= (4/3) * M * R^2 + (8/3) * M * L^2 + 2M * LR
So, the moment of inertia of the assembly about an axis perpendicular to the rod and passing through the rod's other end is (4/3) * M * R^2 + (8/3) * M * L^2 + 2M * LR.
To find the moment of inertia of the assembly, we need to consider the individual moments of inertia of the hollow sphere and the solid rod, and then use the parallel axis theorem to find the total moment of inertia.
1. Moment of inertia of the hollow sphere:
The moment of inertia of a hollow sphere about an axis passing through its center is given by the formula: I_sphere = (2/3) * M * R^2. This formula can be derived using calculus or known mathematical properties of solid objects. Since the mass of the sphere is M and its radius is R, we can find the moment of inertia of the hollow sphere using this formula.
2. Moment of inertia of the solid rod:
The moment of inertia of a solid rod about an axis passing through one end and perpendicular to its length is given by the formula: I_rod = (1/3) * M * L^2. This formula can also be derived using calculus or known mathematical properties of solid objects. Since the mass of the rod is also M and its length is L, we can find the moment of inertia of the solid rod using this formula.
3. Applying the parallel axis theorem:
The parallel axis theorem states that if the moment of inertia of an object about an axis passing through its center of mass is known, the moment of inertia about a parallel axis can be found by adding the mass of the object times the square of the distance between the two axes. In this case, we need to find the moment of inertia of the assembly about an axis perpendicular to the rod and passing through the rod's other end.
Let's denote this perpendicular axis as Axis P and the axis passing through the center of mass of the lollipop assembly as Axis C. The distance between Axis P and Axis C is L, the length of the rod. Since the masses of the hollow sphere and the solid rod are equal and the distances from their centers of mass to the axis of rotation (Axis C) are also equal (R), we can use the parallel axis theorem to find the total moment of inertia (I_total) of the assembly:
I_total = I_sphere + I_rod + M * L^2
Substituting the formulas for I_sphere and I_rod, we get:
I_total = (2/3) * M * R^2 + (1/3) * M * L^2 + M * L^2
Simplifying further:
I_total = (2/3) * M * R^2 + (4/3) * M * L^2
Hence, the moment of inertia of the assembly about the perpendicular axis passing through the rod's other end is (2/3) * M * R^2 + (4/3) * M * L^2.