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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?
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A ring of mass M and radius R is rotating with angular speed a about a...
Analysis of the Problem:
We are given a ring of mass M and radius R rotating with angular speed ω about a fixed vertical axis passing through its center O. There are two point masses, each of mass m, at rest at O. These masses can move radially outwards along two massless rods fixed on the ring. At a certain instant, the angular speed of the system is ω and one of the masses is at a distance of R from O. We need to determine the distance of the other mass from O at this instant.
Solution:
Let's solve this problem step by step.
Step 1: Understanding the Initial Conditions
At the instant when one of the masses is at a distance of R from O, the other mass is at the center O. Since the system is rotating, the angular speed remains constant.
Step 2: Conservation of Angular Momentum
Since there are no external torques acting on the system, the angular momentum of the system is conserved. The total angular momentum of the system is given by:
L = Iω
Where L is the angular momentum, I is the moment of inertia, and ω is the angular speed.
Step 3: Calculating the Moment of Inertia
The moment of inertia of the ring about its axis of rotation passing through its center O is given by:
I_ring = (1/2)MR^2
The moment of inertia of a point mass m at a distance R from the axis of rotation passing through O is given by:
I_point_mass = mR^2
Since there are two point masses, the total moment of inertia of the system is:
I_total = I_ring + 2I_point_mass
= (1/2)MR^2 + 2mR^2
= (1/2)MR^2 + 2mR^2
= (M/2 + 2m)R^2
Step 4: Applying Conservation of Angular Momentum
Using conservation of angular momentum, we can equate the initial and final angular momenta:
I_initial * ω_initial = I_final * ω_final
Since the angular speed remains constant, we have:
I_initial = I_final
Substituting the values of I_initial and I_final, we get:
(M/2 + 2m)R^2 * ω_initial = (M/2 + 2m)R^2 * ω_final
Cancelling out the common terms, we have:
ω_initial = ω_final
Step 5: Finding the Distance of the Other Mass from O
Since the angular speed remains constant, the masses move radially outwards along the rods. Therefore, the distance of the other mass from O remains the same as its initial distance, which is R.
Thus, the distance of the other mass from O at this instant is R.
Final Answer:
The distance of the other mass from O at the instant when one of the masses is at a distance of R from O is R.
We are given a ring of mass M and radius R rotating with angular speed ω about a fixed vertical axis passing through its center O. There are two point masses, each of mass m, at rest at O. These masses can move radially outwards along two massless rods fixed on the ring. At a certain instant, the angular speed of the system is ω and one of the masses is at a distance of R from O. We need to determine the distance of the other mass from O at this instant.
Solution:
Let's solve this problem step by step.
Step 1: Understanding the Initial Conditions
At the instant when one of the masses is at a distance of R from O, the other mass is at the center O. Since the system is rotating, the angular speed remains constant.
Step 2: Conservation of Angular Momentum
Since there are no external torques acting on the system, the angular momentum of the system is conserved. The total angular momentum of the system is given by:
L = Iω
Where L is the angular momentum, I is the moment of inertia, and ω is the angular speed.
Step 3: Calculating the Moment of Inertia
The moment of inertia of the ring about its axis of rotation passing through its center O is given by:
I_ring = (1/2)MR^2
The moment of inertia of a point mass m at a distance R from the axis of rotation passing through O is given by:
I_point_mass = mR^2
Since there are two point masses, the total moment of inertia of the system is:
I_total = I_ring + 2I_point_mass
= (1/2)MR^2 + 2mR^2
= (1/2)MR^2 + 2mR^2
= (M/2 + 2m)R^2
Step 4: Applying Conservation of Angular Momentum
Using conservation of angular momentum, we can equate the initial and final angular momenta:
I_initial * ω_initial = I_final * ω_final
Since the angular speed remains constant, we have:
I_initial = I_final
Substituting the values of I_initial and I_final, we get:
(M/2 + 2m)R^2 * ω_initial = (M/2 + 2m)R^2 * ω_final
Cancelling out the common terms, we have:
ω_initial = ω_final
Step 5: Finding the Distance of the Other Mass from O
Since the angular speed remains constant, the masses move radially outwards along the rods. Therefore, the distance of the other mass from O remains the same as its initial distance, which is R.
Thus, the distance of the other mass from O at this instant is R.
Final Answer:
The distance of the other mass from O at the instant when one of the masses is at a distance of R from O is R.
Community Answer
A ring of mass M and radius R is rotating with angular speed a about a...
Numerical values and diagram is not given in the question but one thing is sure that, you are supposed to apply momentum conservation.
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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?
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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? 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? 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 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? 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? 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?.
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