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A thin circular ring of mass M and radius R is rotating in a horizontal plane about an axis vertical to its plane with a constant angular velocity(ω).if two objects each of mass m be attached gently to the opposite ends of a diameter of the ring, the ring will then rotate with an angular velocity?
Most Upvoted Answer
A thin circular ring of mass M and radius R is rotating in a horizonta...
Let the final angular velocity be w
′
.
Angular momentum is conserved about center of the ring.
L i=L f
Iw+0=Iw′ +2×mr2 w′
Mr 2w+0=Mr2 w ′ +2×mr 2w ′
⟹w ′ = Mw/M+2m.
′
.
Angular momentum is conserved about center of the ring.
L i=L f
Iw+0=Iw′ +2×mr2 w′
Mr 2w+0=Mr2 w ′ +2×mr 2w ′
⟹w ′ = Mw/M+2m.
Community Answer
A thin circular ring of mass M and radius R is rotating in a horizonta...
Problem Analysis:
We are given a thin circular ring of mass M and radius R rotating in a horizontal plane about a vertical axis with a constant angular velocity (ω). Two objects each of mass m are attached gently to the opposite ends of a diameter of the ring. We need to determine the new angular velocity of the ring after the objects are attached.
Solution:
Let's analyze the situation step by step.
Step 1: Initial Angular Momentum:
Before the objects are attached, the ring is rotating with an angular velocity (ω). The angular momentum of the ring can be calculated as the product of its moment of inertia and angular velocity.
Step 2: Conservation of Angular Momentum:
When the objects are attached, the moment of inertia of the system changes. The moment of inertia of the ring-object system can be calculated as the sum of the moment of inertia of the ring and the moment of inertia of the objects.
Step 3: Calculation of Moment of Inertia:
The moment of inertia of the ring is given by the formula:
I_ring = (1/2) * M * R^2
The moment of inertia of each object, considering they are point masses, is given by the formula:
I_object = m * r^2
where r is the distance of the object from the axis of rotation, which is half the radius of the ring (R/2).
Step 4: Final Angular Momentum:
After the objects are attached, the moment of inertia of the system changes. The new angular velocity of the ring can be calculated by equating the initial and final angular momenta.
Conclusion:
The ring will rotate with a new angular velocity given by the equation:
ω_new = (M * ω) / (M + m)
Note:
The above equation assumes that the objects are attached gently, which means there is no external torque acting on the system. This allows for the conservation of angular momentum.
We are given a thin circular ring of mass M and radius R rotating in a horizontal plane about a vertical axis with a constant angular velocity (ω). Two objects each of mass m are attached gently to the opposite ends of a diameter of the ring. We need to determine the new angular velocity of the ring after the objects are attached.
Solution:
Let's analyze the situation step by step.
Step 1: Initial Angular Momentum:
Before the objects are attached, the ring is rotating with an angular velocity (ω). The angular momentum of the ring can be calculated as the product of its moment of inertia and angular velocity.
Step 2: Conservation of Angular Momentum:
When the objects are attached, the moment of inertia of the system changes. The moment of inertia of the ring-object system can be calculated as the sum of the moment of inertia of the ring and the moment of inertia of the objects.
Step 3: Calculation of Moment of Inertia:
The moment of inertia of the ring is given by the formula:
I_ring = (1/2) * M * R^2
The moment of inertia of each object, considering they are point masses, is given by the formula:
I_object = m * r^2
where r is the distance of the object from the axis of rotation, which is half the radius of the ring (R/2).
Step 4: Final Angular Momentum:
After the objects are attached, the moment of inertia of the system changes. The new angular velocity of the ring can be calculated by equating the initial and final angular momenta.
Conclusion:
The ring will rotate with a new angular velocity given by the equation:
ω_new = (M * ω) / (M + m)
Note:
The above equation assumes that the objects are attached gently, which means there is no external torque acting on the system. This allows for the conservation of angular momentum.
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A thin circular ring of mass M and radius R is rotating in a horizontal plane about an axis vertical to its plane with a constant angular velocity(ω).if two objects each of mass m be attached gently to the opposite ends of a diameter of the ring, the ring will then rotate with an angular velocity?
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A thin circular ring of mass M and radius R is rotating in a horizontal plane about an axis vertical to its plane with a constant angular velocity(ω).if two objects each of mass m be attached gently to the opposite ends of a diameter of the ring, the ring will then rotate with an angular velocity? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about A thin circular ring of mass M and radius R is rotating in a horizontal plane about an axis vertical to its plane with a constant angular velocity(ω).if two objects each of mass m be attached gently to the opposite ends of a diameter of the ring, the ring will then rotate with an angular velocity? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A thin circular ring of mass M and radius R is rotating in a horizontal plane about an axis vertical to its plane with a constant angular velocity(ω).if two objects each of mass m be attached gently to the opposite ends of a diameter of the ring, the ring will then rotate with an angular velocity?.
A thin circular ring of mass M and radius R is rotating in a horizontal plane about an axis vertical to its plane with a constant angular velocity(ω).if two objects each of mass m be attached gently to the opposite ends of a diameter of the ring, the ring will then rotate with an angular velocity? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about A thin circular ring of mass M and radius R is rotating in a horizontal plane about an axis vertical to its plane with a constant angular velocity(ω).if two objects each of mass m be attached gently to the opposite ends of a diameter of the ring, the ring will then rotate with an angular velocity? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A thin circular ring of mass M and radius R is rotating in a horizontal plane about an axis vertical to its plane with a constant angular velocity(ω).if two objects each of mass m be attached gently to the opposite ends of a diameter of the ring, the ring will then rotate with an angular velocity?.
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