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What is the mass MOI of right circular cone of radius R and height H about its axis?
- a)4MR2/10
- b)MR2/10
- c)3MR2/10
- d)MR2/12
Correct answer is option 'C'. Can you explain this answer?
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What is the mass MOI of right circular cone of radius R and height H a...
The mass MOI of right circular cone of radius R and height H about its axis is 3MR2/10.
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What is the mass MOI of right circular cone of radius R and height H a...
Understanding the Mass Moment of Inertia (MOI)
The mass moment of inertia (MOI) of an object quantifies its resistance to rotational motion about an axis. For a right circular cone with height H and radius R, we need to derive the MOI about its central axis.
Mass Distribution of a Cone
- A right circular cone can be visualized as a collection of infinitesimally thin discs stacked along its height.
- The mass (M) of the cone is uniformly distributed, and the volume (V) can be calculated as V = (1/3)πR²H.
Deriving the MOI for the Cone
1. Elemental Disc: Consider a thin disc at a height 'z' from the base with thickness 'dz'. Its radius can be expressed as (R/H)z.
2. Volume of the Disc: The volume of this elemental disc is dV = π[(R/H)z]²dz = (πR²z²/H²)dz.
3. Mass of the Disc: The mass of the elemental disc is dm = (M/V)dV = (3M/(πR²H))(πR²z²/H²)dz = (3Mz²/(H³))dz.
4. MOI Contribution: The contribution to the MOI from this disc is dI = (1/2)r²dm = (1/2)((R/H)z)²(3Mz²/(H³))dz.
5. Integrating: The total MOI I is found by integrating from z=0 to z=H.
6. Final Result: After performing the integration, we obtain I = (3/10)MR².
Conclusion
Thus, the mass moment of inertia of a right circular cone about its axis is:
- I = (3/10)MR²
The correct answer is option 'C' - 3MR²/10.
The mass moment of inertia (MOI) of an object quantifies its resistance to rotational motion about an axis. For a right circular cone with height H and radius R, we need to derive the MOI about its central axis.
Mass Distribution of a Cone
- A right circular cone can be visualized as a collection of infinitesimally thin discs stacked along its height.
- The mass (M) of the cone is uniformly distributed, and the volume (V) can be calculated as V = (1/3)πR²H.
Deriving the MOI for the Cone
1. Elemental Disc: Consider a thin disc at a height 'z' from the base with thickness 'dz'. Its radius can be expressed as (R/H)z.
2. Volume of the Disc: The volume of this elemental disc is dV = π[(R/H)z]²dz = (πR²z²/H²)dz.
3. Mass of the Disc: The mass of the elemental disc is dm = (M/V)dV = (3M/(πR²H))(πR²z²/H²)dz = (3Mz²/(H³))dz.
4. MOI Contribution: The contribution to the MOI from this disc is dI = (1/2)r²dm = (1/2)((R/H)z)²(3Mz²/(H³))dz.
5. Integrating: The total MOI I is found by integrating from z=0 to z=H.
6. Final Result: After performing the integration, we obtain I = (3/10)MR².
Conclusion
Thus, the mass moment of inertia of a right circular cone about its axis is:
- I = (3/10)MR²
The correct answer is option 'C' - 3MR²/10.
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What is the mass MOI of right circular cone of radius R and height H about its axis?a)4MR2/10b)MR2/10c)3MR2/10d)MR2/12Correct answer is option 'C'. 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 What is the mass MOI of right circular cone of radius R and height H about its axis?a)4MR2/10b)MR2/10c)3MR2/10d)MR2/12Correct answer is option 'C'. 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 What is the mass MOI of right circular cone of radius R and height H about its axis?a)4MR2/10b)MR2/10c)3MR2/10d)MR2/12Correct answer is option 'C'. Can you explain this answer?.
What is the mass MOI of right circular cone of radius R and height H about its axis?a)4MR2/10b)MR2/10c)3MR2/10d)MR2/12Correct answer is option 'C'. 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 What is the mass MOI of right circular cone of radius R and height H about its axis?a)4MR2/10b)MR2/10c)3MR2/10d)MR2/12Correct answer is option 'C'. 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 What is the mass MOI of right circular cone of radius R and height H about its axis?a)4MR2/10b)MR2/10c)3MR2/10d)MR2/12Correct answer is option 'C'. Can you explain this answer?.
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