Mechanical Engineering Exam > Mechanical Engineering Questions > A thin disc and a thin ring, both have mass M...
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A thin disc and a thin ring, both have mass M and radius R. Both rotate about axes through their centre of mass and are perpendicular to their surfaces at the same angular velocity. Which of the following is true?
- a)The ring has higher kinetic energy
- b)The disc has higher kinetic energy
- c)The ring and the disc have the same kinetic energy
- d)Kinetic energies of both the bodies are zero since they are not in linear motion
Correct answer is option 'A'. Can you explain this answer?
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A thin disc and a thin ring, both have mass M and radius R. Both rotat...
Moment of inertia:
- The moment of inertia of a rigid body about a fixed axis is defined as the sum of the product of the masses of the particles constituting the body and the square of their respective distances from the axis of the rotation.
- The moment of inertia of a particle is
⇒ I = mr2
Where r = the perpendicular distance of the particle from the rotational axis.
Where r = the perpendicular distance of the particle from the rotational axis.
- Moment of inertia of a body made up of a number of particles (discrete distribution)
Rotational kinetic energy:
- The energy, which a body has by virtue of its rotational motion, is called rotational kinetic energy.
- A body rotating about a fixed axis possesses kinetic energy because its constituent particles are in motion, even though the body as a whole remains in place.
- Mathematically rotational kinetic energy can be written as -
Where I = moment of inertia and ω = angular velocity.
Explanation:
- The moment of inertia of the ring about an axis passing through the center and perpendicular to its plane is given by
⇒ Iring = MR2 - Moment of inertia of the disc about an axis passing through center and perpendicular to its plane is given by -
- As we know that mathematically rotational kinetic energy can be written as
- According to the question, the angular velocity of a thin disc and a thin ring is the same. Therefore, the kinetic energy depends on the moment of inertia.
- Therefore, a body having more moments of inertia will have more kinetic energy and vice - versa.
- So, from the equation, it is clear that,
- The ring has higher kinetic energy.
Important Point
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A thin disc and a thin ring, both have mass M and radius R. Both rotat...
Explanation:
When a disc and a ring rotate about axes passing through their centers of mass and perpendicular to their surfaces, they have the same angular velocity. However, their moments of inertia are different due to their different shapes.
Moment of Inertia:
The moment of inertia of an object is a measure of its resistance to rotational motion about a particular axis. It depends on the mass distribution of the object and the axis of rotation.
The moment of inertia for a thin disc rotating about an axis through its center is given by the formula:
I_disc = (1/2) * MR^2
The moment of inertia for a thin ring rotating about an axis through its center is given by the formula:
I_ring = MR^2
Kinetic Energy:
Kinetic energy is the energy an object possesses due to its motion. The kinetic energy of a rotating object depends on its moment of inertia and its angular velocity.
The kinetic energy of a rotating object is given by the formula:
KE = (1/2) * I * ω^2
where KE is the kinetic energy, I is the moment of inertia, and ω is the angular velocity.
Comparison:
Now, let's compare the kinetic energies of the disc and the ring using the formulas above.
The kinetic energy of the disc is given by:
KE_disc = (1/2) * (1/2) * MR^2 * ω^2 = (1/4) * MR^2 * ω^2
The kinetic energy of the ring is given by:
KE_ring = (1/2) * MR^2 * ω^2
Comparing the two expressions, we can see that the kinetic energy of the disc is (1/4) times the kinetic energy of the ring. Therefore, option 'A' is correct - the ring has a higher kinetic energy than the disc.
Summary:
In summary, when a thin disc and a thin ring with the same mass and radius rotate about axes passing through their centers of mass and perpendicular to their surfaces at the same angular velocity, the ring has a higher kinetic energy than the disc. This is due to the difference in their moments of inertia, with the moment of inertia of the disc being half that of the ring.
When a disc and a ring rotate about axes passing through their centers of mass and perpendicular to their surfaces, they have the same angular velocity. However, their moments of inertia are different due to their different shapes.
Moment of Inertia:
The moment of inertia of an object is a measure of its resistance to rotational motion about a particular axis. It depends on the mass distribution of the object and the axis of rotation.
The moment of inertia for a thin disc rotating about an axis through its center is given by the formula:
I_disc = (1/2) * MR^2
The moment of inertia for a thin ring rotating about an axis through its center is given by the formula:
I_ring = MR^2
Kinetic Energy:
Kinetic energy is the energy an object possesses due to its motion. The kinetic energy of a rotating object depends on its moment of inertia and its angular velocity.
The kinetic energy of a rotating object is given by the formula:
KE = (1/2) * I * ω^2
where KE is the kinetic energy, I is the moment of inertia, and ω is the angular velocity.
Comparison:
Now, let's compare the kinetic energies of the disc and the ring using the formulas above.
The kinetic energy of the disc is given by:
KE_disc = (1/2) * (1/2) * MR^2 * ω^2 = (1/4) * MR^2 * ω^2
The kinetic energy of the ring is given by:
KE_ring = (1/2) * MR^2 * ω^2
Comparing the two expressions, we can see that the kinetic energy of the disc is (1/4) times the kinetic energy of the ring. Therefore, option 'A' is correct - the ring has a higher kinetic energy than the disc.
Summary:
In summary, when a thin disc and a thin ring with the same mass and radius rotate about axes passing through their centers of mass and perpendicular to their surfaces at the same angular velocity, the ring has a higher kinetic energy than the disc. This is due to the difference in their moments of inertia, with the moment of inertia of the disc being half that of the ring.
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A thin disc and a thin ring, both have mass M and radius R. Both rotate about axes through their centre of mass and are perpendicular to their surfaces at the same angular velocity. Which of the following is true?a)The ring has higher kinetic energyb)The disc has higher kinetic energyc)The ring and the disc have the same kinetic energyd)Kinetic energies of both the bodies are zero since they are not in linear motionCorrect answer is option 'A'. Can you explain this answer?
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A thin disc and a thin ring, both have mass M and radius R. Both rotate about axes through their centre of mass and are perpendicular to their surfaces at the same angular velocity. Which of the following is true?a)The ring has higher kinetic energyb)The disc has higher kinetic energyc)The ring and the disc have the same kinetic energyd)Kinetic energies of both the bodies are zero since they are not in linear motionCorrect answer is option 'A'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about A thin disc and a thin ring, both have mass M and radius R. Both rotate about axes through their centre of mass and are perpendicular to their surfaces at the same angular velocity. Which of the following is true?a)The ring has higher kinetic energyb)The disc has higher kinetic energyc)The ring and the disc have the same kinetic energyd)Kinetic energies of both the bodies are zero since they are not in linear motionCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A thin disc and a thin ring, both have mass M and radius R. Both rotate about axes through their centre of mass and are perpendicular to their surfaces at the same angular velocity. Which of the following is true?a)The ring has higher kinetic energyb)The disc has higher kinetic energyc)The ring and the disc have the same kinetic energyd)Kinetic energies of both the bodies are zero since they are not in linear motionCorrect answer is option 'A'. Can you explain this answer?.
A thin disc and a thin ring, both have mass M and radius R. Both rotate about axes through their centre of mass and are perpendicular to their surfaces at the same angular velocity. Which of the following is true?a)The ring has higher kinetic energyb)The disc has higher kinetic energyc)The ring and the disc have the same kinetic energyd)Kinetic energies of both the bodies are zero since they are not in linear motionCorrect answer is option 'A'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about A thin disc and a thin ring, both have mass M and radius R. Both rotate about axes through their centre of mass and are perpendicular to their surfaces at the same angular velocity. Which of the following is true?a)The ring has higher kinetic energyb)The disc has higher kinetic energyc)The ring and the disc have the same kinetic energyd)Kinetic energies of both the bodies are zero since they are not in linear motionCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A thin disc and a thin ring, both have mass M and radius R. Both rotate about axes through their centre of mass and are perpendicular to their surfaces at the same angular velocity. Which of the following is true?a)The ring has higher kinetic energyb)The disc has higher kinetic energyc)The ring and the disc have the same kinetic energyd)Kinetic energies of both the bodies are zero since they are not in linear motionCorrect answer is option 'A'. Can you explain this answer?.
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A thin disc and a thin ring, both have mass M and radius R. Both rotate about axes through their centre of mass and are perpendicular to their surfaces at the same angular velocity. Which of the following is true?a)The ring has higher kinetic energyb)The disc has higher kinetic energyc)The ring and the disc have the same kinetic energyd)Kinetic energies of both the bodies are zero since they are not in linear motionCorrect answer is option 'A'. Can you explain this answer?, a detailed solution for A thin disc and a thin ring, both have mass M and radius R. Both rotate about axes through their centre of mass and are perpendicular to their surfaces at the same angular velocity. Which of the following is true?a)The ring has higher kinetic energyb)The disc has higher kinetic energyc)The ring and the disc have the same kinetic energyd)Kinetic energies of both the bodies are zero since they are not in linear motionCorrect answer is option 'A'. Can you explain this answer? has been provided alongside types of A thin disc and a thin ring, both have mass M and radius R. Both rotate about axes through their centre of mass and are perpendicular to their surfaces at the same angular velocity. Which of the following is true?a)The ring has higher kinetic energyb)The disc has higher kinetic energyc)The ring and the disc have the same kinetic energyd)Kinetic energies of both the bodies are zero since they are not in linear motionCorrect answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
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