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Moment of inertia of a circular wire of mass M and radius R about its diameter is _______
- a)1/2 MR2
- b)1/4 MR2
- c)2MR2
- d)MR2
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
Moment of inertia of a circular wire of mass M and radius R about its ...
A circular wire behaves like a ring. By perpendicular axes theorem,
ID + ID = 1/2 MR2
Therefore,
1/4 MR2.
ID + ID = 1/2 MR2
Therefore,
1/4 MR2.
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Moment of inertia of a circular wire of mass M and radius R about its ...
Understanding Moment of Inertia
The moment of inertia (I) is a measure of an object's resistance to angular acceleration about a particular axis. For a circular wire, it depends on both its mass and the distribution of that mass relative to the axis of rotation.
Moment of Inertia of a Circular Wire
To find the moment of inertia of a circular wire about its diameter, we can use principles of integration or known results from physics.
Key Concepts
- Mass Distribution: The circular wire has a uniform mass distribution along its length.
- Axis of Rotation: We are interested in the axis that runs through the diameter of the circle.
Calculation Details
- The moment of inertia for a thin circular wire about an axis through its center and perpendicular to its plane (the z-axis) is given by the formula I = MR^2.
- By using the perpendicular axis theorem, we can derive the moment of inertia about the diameter (x or y-axis). The perpendicular axis theorem states that for a planar body:
I_z = I_x + I_y, where I_z is the moment of inertia about the axis perpendicular to the plane, and I_x and I_y are about the two perpendicular axes in the plane.
- For a circular wire, I_z = MR^2. Since I_x = I_y for symmetry, we have:
I_z = I_x + I_y = 2I_x.
Thus, I_x = I_z / 2 = MR^2 / 2.
Conclusion
Therefore, the moment of inertia of a circular wire of mass M and radius R about its diameter is:
Answer: I = 1/2 MR^2
This confirms that option 'B' is correct.
The moment of inertia (I) is a measure of an object's resistance to angular acceleration about a particular axis. For a circular wire, it depends on both its mass and the distribution of that mass relative to the axis of rotation.
Moment of Inertia of a Circular Wire
To find the moment of inertia of a circular wire about its diameter, we can use principles of integration or known results from physics.
Key Concepts
- Mass Distribution: The circular wire has a uniform mass distribution along its length.
- Axis of Rotation: We are interested in the axis that runs through the diameter of the circle.
Calculation Details
- The moment of inertia for a thin circular wire about an axis through its center and perpendicular to its plane (the z-axis) is given by the formula I = MR^2.
- By using the perpendicular axis theorem, we can derive the moment of inertia about the diameter (x or y-axis). The perpendicular axis theorem states that for a planar body:
I_z = I_x + I_y, where I_z is the moment of inertia about the axis perpendicular to the plane, and I_x and I_y are about the two perpendicular axes in the plane.
- For a circular wire, I_z = MR^2. Since I_x = I_y for symmetry, we have:
I_z = I_x + I_y = 2I_x.
Thus, I_x = I_z / 2 = MR^2 / 2.
Conclusion
Therefore, the moment of inertia of a circular wire of mass M and radius R about its diameter is:
Answer: I = 1/2 MR^2
This confirms that option 'B' is correct.
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Moment of inertia of a circular wire of mass M and radius R about its diameter is _______a)1/2 MR2b)1/4 MR2c)2MR2d)MR2Correct answer is option 'B'. Can you explain this answer? for Mechanical Engineering 2025 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about Moment of inertia of a circular wire of mass M and radius R about its diameter is _______a)1/2 MR2b)1/4 MR2c)2MR2d)MR2Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Moment of inertia of a circular wire of mass M and radius R about its diameter is _______a)1/2 MR2b)1/4 MR2c)2MR2d)MR2Correct answer is option 'B'. Can you explain this answer?.
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