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Two bodies, a ring and a solid cylinder of same material are rolling down without slipping an inclined plane. The radii of the bodies are same. The ratio of velocity of the centre of mass at the bottom of the inclined plane of the ring to that of the cylinder is √x/2. Then the value of x is _______.
Correct answer is '3'. Can you explain this answer?
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Two bodies, a ring and a solid cylinder of same material are rolling d...
To solve this problem, we need to use the concept of rotational and translational kinetic energy.
Let's denote the mass of the ring as m and its radius as R.
The moment of inertia of the ring about its center of mass is I = mR^2.
Let's denote the mass of the cylinder as M and its radius as R.
The moment of inertia of the cylinder about its center of mass is I = 1/2MR^2.
When the two bodies roll down the inclined plane without slipping, their rotational kinetic energy is converted into translational kinetic energy of the center of mass.
The translational kinetic energy of the ring is given by:
KE_ring = 1/2mv^2, where v is the velocity of the center of mass of the ring at the bottom of the inclined plane.
The translational kinetic energy of the cylinder is given by:
KE_cylinder = 1/2MV^2, where V is the velocity of the center of mass of the cylinder at the bottom of the inclined plane.
The total mechanical energy of the system is conserved, so we have:
KE_ring + KE_cylinder = constant.
Since the radii of the ring and the cylinder are the same, the center of mass of both bodies will have the same vertical displacement when they reach the bottom of the inclined plane. Let's denote this vertical displacement as h.
At the bottom of the inclined plane, the gravitational potential energy of both bodies is converted into kinetic energy, so we have:
mgh + Mgh = KE_ring + KE_cylinder.
Since the total mechanical energy is conserved, we can write:
mgh + Mgh = 1/2mv^2 + 1/2MV^2.
Since the mass of the ring and the cylinder are the same, we can cancel out the mass term:
gh + gh = 1/2v^2 + 1/2V^2.
Dividing both sides by gh, we get:
2 = v^2/V^2.
Taking the square root of both sides, we get:
√2 = v/V.
Therefore, the ratio of the velocity of the center of mass at the bottom of the inclined plane of the ring to that of the cylinder is √2.
Let's denote the mass of the ring as m and its radius as R.
The moment of inertia of the ring about its center of mass is I = mR^2.
Let's denote the mass of the cylinder as M and its radius as R.
The moment of inertia of the cylinder about its center of mass is I = 1/2MR^2.
When the two bodies roll down the inclined plane without slipping, their rotational kinetic energy is converted into translational kinetic energy of the center of mass.
The translational kinetic energy of the ring is given by:
KE_ring = 1/2mv^2, where v is the velocity of the center of mass of the ring at the bottom of the inclined plane.
The translational kinetic energy of the cylinder is given by:
KE_cylinder = 1/2MV^2, where V is the velocity of the center of mass of the cylinder at the bottom of the inclined plane.
The total mechanical energy of the system is conserved, so we have:
KE_ring + KE_cylinder = constant.
Since the radii of the ring and the cylinder are the same, the center of mass of both bodies will have the same vertical displacement when they reach the bottom of the inclined plane. Let's denote this vertical displacement as h.
At the bottom of the inclined plane, the gravitational potential energy of both bodies is converted into kinetic energy, so we have:
mgh + Mgh = KE_ring + KE_cylinder.
Since the total mechanical energy is conserved, we can write:
mgh + Mgh = 1/2mv^2 + 1/2MV^2.
Since the mass of the ring and the cylinder are the same, we can cancel out the mass term:
gh + gh = 1/2v^2 + 1/2V^2.
Dividing both sides by gh, we get:
2 = v^2/V^2.
Taking the square root of both sides, we get:
√2 = v/V.
Therefore, the ratio of the velocity of the center of mass at the bottom of the inclined plane of the ring to that of the cylinder is √2.
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Community Answer
Two bodies, a ring and a solid cylinder of same material are rolling d...
I in both cases is about point of contact.
Ring:
Ring:
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Two bodies, a ring and a solid cylinder of same material are rolling down without slipping an inclined plane. The radii of the bodies are same. The ratio of velocity of the centre of mass at the bottom of the inclined plane of the ring to that of the cylinder is √x/2. Then the value of x is _______.Correct answer is '3'. Can you explain this answer?
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Two bodies, a ring and a solid cylinder of same material are rolling down without slipping an inclined plane. The radii of the bodies are same. The ratio of velocity of the centre of mass at the bottom of the inclined plane of the ring to that of the cylinder is √x/2. Then the value of x is _______.Correct answer is '3'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Two bodies, a ring and a solid cylinder of same material are rolling down without slipping an inclined plane. The radii of the bodies are same. The ratio of velocity of the centre of mass at the bottom of the inclined plane of the ring to that of the cylinder is √x/2. Then the value of x is _______.Correct answer is '3'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two bodies, a ring and a solid cylinder of same material are rolling down without slipping an inclined plane. The radii of the bodies are same. The ratio of velocity of the centre of mass at the bottom of the inclined plane of the ring to that of the cylinder is √x/2. Then the value of x is _______.Correct answer is '3'. Can you explain this answer?.
Two bodies, a ring and a solid cylinder of same material are rolling down without slipping an inclined plane. The radii of the bodies are same. The ratio of velocity of the centre of mass at the bottom of the inclined plane of the ring to that of the cylinder is √x/2. Then the value of x is _______.Correct answer is '3'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Two bodies, a ring and a solid cylinder of same material are rolling down without slipping an inclined plane. The radii of the bodies are same. The ratio of velocity of the centre of mass at the bottom of the inclined plane of the ring to that of the cylinder is √x/2. Then the value of x is _______.Correct answer is '3'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two bodies, a ring and a solid cylinder of same material are rolling down without slipping an inclined plane. The radii of the bodies are same. The ratio of velocity of the centre of mass at the bottom of the inclined plane of the ring to that of the cylinder is √x/2. Then the value of x is _______.Correct answer is '3'. Can you explain this answer?.
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Two bodies, a ring and a solid cylinder of same material are rolling down without slipping an inclined plane. The radii of the bodies are same. The ratio of velocity of the centre of mass at the bottom of the inclined plane of the ring to that of the cylinder is √x/2. Then the value of x is _______.Correct answer is '3'. Can you explain this answer?, a detailed solution for Two bodies, a ring and a solid cylinder of same material are rolling down without slipping an inclined plane. The radii of the bodies are same. The ratio of velocity of the centre of mass at the bottom of the inclined plane of the ring to that of the cylinder is √x/2. Then the value of x is _______.Correct answer is '3'. Can you explain this answer? has been provided alongside types of Two bodies, a ring and a solid cylinder of same material are rolling down without slipping an inclined plane. The radii of the bodies are same. The ratio of velocity of the centre of mass at the bottom of the inclined plane of the ring to that of the cylinder is √x/2. Then the value of x is _______.Correct answer is '3'. Can you explain this answer? theory, EduRev gives you an
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