What is the moment of inertia of a thin wire bent in the shape of a semicircle o

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Question: What is the moment of inertia of a thin wire bent in the shape of a semicircle of radius R and mass M about the diameter?

Options:

Correct Answer: 3/8 MR^2

Solution:

The moment of inertia of a thin wire bent in the shape of a semicircle about the diameter is I = 3/8 MR^2.

What is the moment of inertia of a thin wire bent in the shape of a semicircle of radius R and mass M about the diameter?

What is the moment of inertia of a thin wire bent in the shape of a semicircle of radius R and mass M about the diameter?

Step 1: Understand the concept of moment of inertia. It measures how difficult it is to rotate an object around an axis.
Step 2: Identify the shape of the object. In this case, it is a thin wire bent into a semicircle.
Step 3: Recognize the axis of rotation. We are calculating the moment of inertia about the diameter of the semicircle.
Step 4: Use the formula for the moment of inertia of a thin wire. For a semicircular wire, the moment of inertia about the diameter is derived from integration.
Step 5: The formula for the moment of inertia of a semicircular wire about its diameter is I = (1/2) * M * R^2, but we need to adjust it for the semicircle shape.
Step 6: After performing the necessary calculations, we find that the moment of inertia of the semicircular wire about the diameter is I = (3/8) * M * R^2.
Step 7: Conclude that the final answer is I = (3/8) * M * R^2.

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