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A ring of mass M and radius R is rotating with angular speed a about a fixed vertical axis passing through its centre O with two point masses each of mass at rest at O. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant the angular speed of the system isand one of the masses is 3 at a distance of R from O. At this instant the distance of the other mass from O ??(a) 2/3R (b) 1/3R (c) 3/5R (d) 4/5R?
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A ring of mass M and radius R is rotating with angular speed a about a...
To solve this problem, let's consider the conservation of angular momentum.
1. Conservation of Angular Momentum:
The angular momentum of a rotating object is given by the equation L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity. In this case, the ring and the two masses contribute to the total angular momentum of the system.
2. Initial Angular Momentum:
At the initial instant, the angular velocity is given as ω. The moment of inertia of the ring is MR²/2, and the moment of inertia of each mass at O is zero since they are at rest. Therefore, the total initial angular momentum is L₀ = (MR²/2)ω.
3. Final Angular Momentum:
At the final instant, one of the masses is at a distance of R from O, which means it has moved radially outwards. The other mass is not mentioned, so we need to find its distance from O.
4. Conservation of Angular Momentum Equation:
According to the principle of conservation of angular momentum, the initial angular momentum L₀ is equal to the final angular momentum L₁. Since the moment of inertia of the ring remains the same and the masses move radially, the moment of inertia of the system remains constant.
5. Deriving the Equation:
L₀ = L₁
(MR²/2)ω = (MR²/2)a + m₁r₁²ω₁ + m₂r₂²ω₂
Since the masses are at rest at O, ω₁ and ω₂ are both zero.
(MR²/2)ω = (MR²/2)a + m₁r₁²(0) + m₂r₂²(0)
(MR²/2)ω = (MR²/2)a
ω = a
6. Solving for the Distance:
ω = a implies that the angular velocity at the final instant is equal to the angular acceleration. Substituting this in the equation above:
(MR²/2)a = (MR²/2)a + m₁r₁²(0) + m₂r₂²(0)
(MR²/2)a = (MR²/2)a
This equation is satisfied for any values of r₁ and r₂ as long as m₁ + m₂ = M.
7. Conclusion:
Since the equation is satisfied for any values of r₁ and r₂, the distance of the other mass from O can be any value as long as m₁ + m₂ = M. Therefore, the distance of the other mass from O cannot be determined based on the given information. The correct answer is (None of the above).
1. Conservation of Angular Momentum:
The angular momentum of a rotating object is given by the equation L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity. In this case, the ring and the two masses contribute to the total angular momentum of the system.
2. Initial Angular Momentum:
At the initial instant, the angular velocity is given as ω. The moment of inertia of the ring is MR²/2, and the moment of inertia of each mass at O is zero since they are at rest. Therefore, the total initial angular momentum is L₀ = (MR²/2)ω.
3. Final Angular Momentum:
At the final instant, one of the masses is at a distance of R from O, which means it has moved radially outwards. The other mass is not mentioned, so we need to find its distance from O.
4. Conservation of Angular Momentum Equation:
According to the principle of conservation of angular momentum, the initial angular momentum L₀ is equal to the final angular momentum L₁. Since the moment of inertia of the ring remains the same and the masses move radially, the moment of inertia of the system remains constant.
5. Deriving the Equation:
L₀ = L₁
(MR²/2)ω = (MR²/2)a + m₁r₁²ω₁ + m₂r₂²ω₂
Since the masses are at rest at O, ω₁ and ω₂ are both zero.
(MR²/2)ω = (MR²/2)a + m₁r₁²(0) + m₂r₂²(0)
(MR²/2)ω = (MR²/2)a
ω = a
6. Solving for the Distance:
ω = a implies that the angular velocity at the final instant is equal to the angular acceleration. Substituting this in the equation above:
(MR²/2)a = (MR²/2)a + m₁r₁²(0) + m₂r₂²(0)
(MR²/2)a = (MR²/2)a
This equation is satisfied for any values of r₁ and r₂ as long as m₁ + m₂ = M.
7. Conclusion:
Since the equation is satisfied for any values of r₁ and r₂, the distance of the other mass from O can be any value as long as m₁ + m₂ = M. Therefore, the distance of the other mass from O cannot be determined based on the given information. The correct answer is (None of the above).
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A ring of mass M and radius R is rotating with angular speed a about a fixed vertical axis passing through its centre O with two point masses each of mass at rest at O. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant the angular speed of the system isand one of the masses is 3 at a distance of R from O. At this instant the distance of the other mass from O ??(a) 2/3R (b) 1/3R (c) 3/5R (d) 4/5R?
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A ring of mass M and radius R is rotating with angular speed a about a fixed vertical axis passing through its centre O with two point masses each of mass at rest at O. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant the angular speed of the system isand one of the masses is 3 at a distance of R from O. At this instant the distance of the other mass from O ??(a) 2/3R (b) 1/3R (c) 3/5R (d) 4/5R? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A ring of mass M and radius R is rotating with angular speed a about a fixed vertical axis passing through its centre O with two point masses each of mass at rest at O. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant the angular speed of the system isand one of the masses is 3 at a distance of R from O. At this instant the distance of the other mass from O ??(a) 2/3R (b) 1/3R (c) 3/5R (d) 4/5R? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A ring of mass M and radius R is rotating with angular speed a about a fixed vertical axis passing through its centre O with two point masses each of mass at rest at O. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant the angular speed of the system isand one of the masses is 3 at a distance of R from O. At this instant the distance of the other mass from O ??(a) 2/3R (b) 1/3R (c) 3/5R (d) 4/5R?.
A ring of mass M and radius R is rotating with angular speed a about a fixed vertical axis passing through its centre O with two point masses each of mass at rest at O. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant the angular speed of the system isand one of the masses is 3 at a distance of R from O. At this instant the distance of the other mass from O ??(a) 2/3R (b) 1/3R (c) 3/5R (d) 4/5R? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A ring of mass M and radius R is rotating with angular speed a about a fixed vertical axis passing through its centre O with two point masses each of mass at rest at O. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant the angular speed of the system isand one of the masses is 3 at a distance of R from O. At this instant the distance of the other mass from O ??(a) 2/3R (b) 1/3R (c) 3/5R (d) 4/5R? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A ring of mass M and radius R is rotating with angular speed a about a fixed vertical axis passing through its centre O with two point masses each of mass at rest at O. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant the angular speed of the system isand one of the masses is 3 at a distance of R from O. At this instant the distance of the other mass from O ??(a) 2/3R (b) 1/3R (c) 3/5R (d) 4/5R?.
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