A circle is cut out of a uniform piece of plywood. How does the rotational inertia measured around a point on its outer edge
compare with the rotational inertia about its central axis?
a. The two rotational inertias are the same.
b. The rotational inertia about its edge is twice as great as it is about its center.
c. The rotational inertia about its edge is three times as great as it is about its center.
d. The rotational inertia about its edge is half as great as it is about its center.
e. The rotational inertia about its edge is one third as great as it is about its center.
Also the reason for it.
The parallel axis theorem applies
Inew=Iold+Md^2 d is the distance axis of rotation moved.
Since I old was 1/2 Mr^2, and now it is Inew=3/2 Mr^2....answer C applies
The correct answer is d. The rotational inertia about the edge is half as great as it is about the center.
To understand why this is the case, it is important to know that rotational inertia (also known as moment of inertia) depends on both the mass distribution and the distance of the mass from the axis of rotation.
When a circle is cut out of a uniform piece of plywood, the resulting shape resembles a hollow cylinder or a ring. Let's compare the rotational inertia of this ring about its outer edge (which is at a maximum distance from the axis of rotation) and its central axis.
The moment of inertia for a thin ring of uniform density is given by the equation:
I = MR²
In this equation, M represents the total mass of the ring and R represents the radius of the ring.
When considering the rotational inertia about the edge, the radius R is the maximum distance from the axis of rotation. On the other hand, when considering the rotational inertia about the central axis, the radius R is zero since it is the distance from the axis to itself.
Since the rotational inertia depends on the square of the radius, when R approaches zero (as in the case of the central axis), the moment of inertia decreases significantly. Therefore, the rotational inertia about the edge is half (or 1/2) as great as it is about the center.
In summary, the rotational inertia measured around a point on the outer edge of a circle (or ring) is half as great as the rotational inertia measured about its central axis.