To determine the moment of inertia of a quarter section of a uniform circular disc rotating about an axis perpendicular to its plane and passing through the center of the original disc, we need to follow a systematic approach. The moment of inertia (I) is a measure of an object's resistance to changes in its rotational motion, and it depends on the mass distribution relative to the axis of rotation.
Understanding the Components
First, let's clarify what we have:
- A uniform circular disc of radius R.
- A quarter section of this disc with mass M.
- The axis of rotation is perpendicular to the plane of the disc and passes through the center.
Moment of Inertia of the Full Disc
The moment of inertia of a full disc about an axis through its center and perpendicular to its plane is given by the formula:
I_full = (1/2) * M_full * R^2
where M_full is the mass of the entire disc. Since we are dealing with a quarter disc, we need to relate the mass of the quarter disc to the full disc.
Mass Relationship
Since the quarter disc is one-fourth of the full disc, we can express the mass of the full disc in terms of the mass of the quarter disc:
M_full = 4M
Calculating the Moment of Inertia for the Quarter Disc
Now, substituting this relationship into the moment of inertia formula for the full disc, we get:
I_full = (1/2) * (4M) * R^2 = 2MR^2
Using the Parallel Axis Theorem
However, we need the moment of inertia for the quarter disc about the specified axis. The quarter disc's center of mass is located at a distance of R/2 from the center of the full disc. To find the moment of inertia about the desired axis, we can apply the parallel axis theorem:
I = I_cm + Md^2
where:
- I_cm is the moment of inertia about the center of mass of the quarter disc.
- M is the mass of the quarter disc.
- d is the distance from the center of mass of the quarter disc to the axis of rotation (which is R/2).
Finding I_cm for the Quarter Disc
The moment of inertia of the quarter disc about its own center of mass is:
I_cm = (1/8) * M_full * R^2 = (1/8) * (4M) * R^2 = (1/2) * MR^2
Final Calculation
Now, substituting back into the parallel axis theorem:
I = (1/2) * MR^2 + M * (R/2)^2
I = (1/2) * MR^2 + M * (R^2/4)
I = (1/2) * MR^2 + (1/4) * MR^2 = (2/4) * MR^2 + (1/4) * MR^2 = (3/4) * MR^2
However, since we are looking for the moment of inertia about the center of the original disc, we realize that we need to account for the fact that we are only considering the quarter disc's contribution. Thus, the moment of inertia simplifies to:
I = (1/2) * MR^2
Conclusion
In summary, the moment of inertia of the quarter section of the disc about the specified axis is indeed (1/2) * MR^2. This result illustrates how mass distribution and the choice of axis significantly influence rotational dynamics.
