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Two cylinders have the same mass and radius. One is hollow and the other is solid. Which one will have the greater moment of inertia about the central axis?
- a)Hollow cylinder
- b)Solid cylinder
- c)Same for both
- d)Depends on length of cylinder
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
Two cylinders have the same mass and radius. One is hollow and the oth...
The hollow one will have greater density as both have the same mass and it has to be distributed uniformly. Thus, the summation of product of mass and perpendicular distance from the axis squared of each particle will be more for the hollow cylinder. And hence, it has more moment of inertia.
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Two cylinders have the same mass and radius. One is hollow and the oth...
Understanding Moment of Inertia
The moment of inertia (I) is a measure of an object's resistance to changes in its rotational motion. It depends on the mass distribution relative to the axis of rotation.
Moment of Inertia Formulas
- For a solid cylinder:
I_solid = (1/2) * M * R^2
- For a hollow cylinder (thin-walled):
I_hollow = M * R^2
Here, M is the mass and R is the radius of the cylinder.
Comparing the Two Cylinders
- Mass and Radius: Both cylinders have the same mass (M) and radius (R).
- Calculation of I:
- For the solid cylinder: I_solid = (1/2) * M * R^2
- For the hollow cylinder: I_hollow = M * R^2
Key Observations
- The moment of inertia of the hollow cylinder is greater than that of the solid cylinder.
- Numerical Comparison:
If we take a mass of 1 kg and a radius of 1 meter:
- I_solid = (1/2) * 1 * (1)^2 = 0.5 kg·m²
- I_hollow = 1 * (1)^2 = 1 kg·m²
Thus, I_hollow (1 kg·m²) > I_solid (0.5 kg·m²).
Conclusion
The hollow cylinder has a greater moment of inertia compared to the solid cylinder due to its mass being distributed further from the central axis. This is why the correct answer is option 'A': Hollow cylinder.
The moment of inertia (I) is a measure of an object's resistance to changes in its rotational motion. It depends on the mass distribution relative to the axis of rotation.
Moment of Inertia Formulas
- For a solid cylinder:
I_solid = (1/2) * M * R^2
- For a hollow cylinder (thin-walled):
I_hollow = M * R^2
Here, M is the mass and R is the radius of the cylinder.
Comparing the Two Cylinders
- Mass and Radius: Both cylinders have the same mass (M) and radius (R).
- Calculation of I:
- For the solid cylinder: I_solid = (1/2) * M * R^2
- For the hollow cylinder: I_hollow = M * R^2
Key Observations
- The moment of inertia of the hollow cylinder is greater than that of the solid cylinder.
- Numerical Comparison:
If we take a mass of 1 kg and a radius of 1 meter:
- I_solid = (1/2) * 1 * (1)^2 = 0.5 kg·m²
- I_hollow = 1 * (1)^2 = 1 kg·m²
Thus, I_hollow (1 kg·m²) > I_solid (0.5 kg·m²).
Conclusion
The hollow cylinder has a greater moment of inertia compared to the solid cylinder due to its mass being distributed further from the central axis. This is why the correct answer is option 'A': Hollow cylinder.
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Two cylinders have the same mass and radius. One is hollow and the other is solid. Which one will have the greater moment of inertia about the central axis?a)Hollow cylinderb)Solid cylinderc)Same for bothd)Depends on length of cylinderCorrect answer is option 'A'. Can you explain this answer?
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