UPSC Exam > UPSC Questions > A solid sphere has density p in the top half ...
Start Learning for Free
A solid sphere has density p in the top half and 2p in the bottom half. it is kept in a uniform fluid of density 3p with a hinge at the bottom such that it can freely perform oscillations about horizontal axis through the hinge ( radius of sphere is R) find period of small oscillations?
Most Upvoted Answer
A solid sphere has density p in the top half and 2p in the bottom half...
Introduction
To determine the period of small oscillations of a solid sphere with varying density, we need to analyze the system's equilibrium and the forces acting upon it. The sphere has density p in the top half and 2p in the bottom half, and it is submerged in a fluid with density 3p.
Density and Center of Mass
- The sphere's total mass can be calculated by integrating the densities:
- Top half mass (p): M1 = (2/3)πR^3p
- Bottom half mass (2p): M2 = (1/3)πR^3(2p) = (2/3)πR^3p
- Total mass (M) = M1 + M2 = (4/3)πR^3p
- To find the center of mass (CM), consider the weighted average:
- CM from the hinge downwards = (1/2)R * (M1 + (3/2)R * M2) / M
Buoyancy Force and Torque
- The buoyant force (B) acting on the sphere is equal to the weight of the fluid displaced:
- B = V_fluid * density_fluid = (4/3)πR^3(3p) = 4πR^3p
- The torque (τ) about the hinge due to the buoyancy when displaced by an angle θ:
- τ = B * (l - h) = 4πR^3p * (R * sin(θ))
Angular Frequency and Period Calculation
- The angular frequency (ω) for small oscillations can be approximated:
- ω^2 = τ/I, where I is the moment of inertia of the sphere (I = 2/5 * M * R^2).
- Period (T) of the oscillation is given by:
- T = 2π/ω
The final expression for T can be derived from the above relationships, leading to a formula that incorporates R, p, and fluid density.
Conclusion
The period of small oscillations of the sphere can then be expressed in terms of its physical properties and densities, illustrating the balance between gravitational and buoyant forces in the system.
To determine the period of small oscillations of a solid sphere with varying density, we need to analyze the system's equilibrium and the forces acting upon it. The sphere has density p in the top half and 2p in the bottom half, and it is submerged in a fluid with density 3p.
Density and Center of Mass
- The sphere's total mass can be calculated by integrating the densities:
- Top half mass (p): M1 = (2/3)πR^3p
- Bottom half mass (2p): M2 = (1/3)πR^3(2p) = (2/3)πR^3p
- Total mass (M) = M1 + M2 = (4/3)πR^3p
- To find the center of mass (CM), consider the weighted average:
- CM from the hinge downwards = (1/2)R * (M1 + (3/2)R * M2) / M
Buoyancy Force and Torque
- The buoyant force (B) acting on the sphere is equal to the weight of the fluid displaced:
- B = V_fluid * density_fluid = (4/3)πR^3(3p) = 4πR^3p
- The torque (τ) about the hinge due to the buoyancy when displaced by an angle θ:
- τ = B * (l - h) = 4πR^3p * (R * sin(θ))
Angular Frequency and Period Calculation
- The angular frequency (ω) for small oscillations can be approximated:
- ω^2 = τ/I, where I is the moment of inertia of the sphere (I = 2/5 * M * R^2).
- Period (T) of the oscillation is given by:
- T = 2π/ω
The final expression for T can be derived from the above relationships, leading to a formula that incorporates R, p, and fluid density.
Conclusion
The period of small oscillations of the sphere can then be expressed in terms of its physical properties and densities, illustrating the balance between gravitational and buoyant forces in the system.
Attention UPSC Students!
To make sure you are not studying endlessly, EduRev has designed UPSC study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in UPSC.
|
Explore Courses for UPSC exam
|
|
Similar UPSC Doubts
Top Courses for UPSCView all
A solid sphere has density p in the top half and 2p in the bottom half. it is kept in a uniform fluid of density 3p with a hinge at the bottom such that it can freely perform oscillations about horizontal axis through the hinge ( radius of sphere is R) find period of small oscillations?
Question Description
A solid sphere has density p in the top half and 2p in the bottom half. it is kept in a uniform fluid of density 3p with a hinge at the bottom such that it can freely perform oscillations about horizontal axis through the hinge ( radius of sphere is R) find period of small oscillations? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A solid sphere has density p in the top half and 2p in the bottom half. it is kept in a uniform fluid of density 3p with a hinge at the bottom such that it can freely perform oscillations about horizontal axis through the hinge ( radius of sphere is R) find period of small oscillations? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A solid sphere has density p in the top half and 2p in the bottom half. it is kept in a uniform fluid of density 3p with a hinge at the bottom such that it can freely perform oscillations about horizontal axis through the hinge ( radius of sphere is R) find period of small oscillations?.
A solid sphere has density p in the top half and 2p in the bottom half. it is kept in a uniform fluid of density 3p with a hinge at the bottom such that it can freely perform oscillations about horizontal axis through the hinge ( radius of sphere is R) find period of small oscillations? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A solid sphere has density p in the top half and 2p in the bottom half. it is kept in a uniform fluid of density 3p with a hinge at the bottom such that it can freely perform oscillations about horizontal axis through the hinge ( radius of sphere is R) find period of small oscillations? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A solid sphere has density p in the top half and 2p in the bottom half. it is kept in a uniform fluid of density 3p with a hinge at the bottom such that it can freely perform oscillations about horizontal axis through the hinge ( radius of sphere is R) find period of small oscillations?.
Solutions for A solid sphere has density p in the top half and 2p in the bottom half. it is kept in a uniform fluid of density 3p with a hinge at the bottom such that it can freely perform oscillations about horizontal axis through the hinge ( radius of sphere is R) find period of small oscillations? in English & in Hindi are available as part of our courses for UPSC.
Download more important topics, notes, lectures and mock test series for UPSC Exam by signing up for free.
Here you can find the meaning of A solid sphere has density p in the top half and 2p in the bottom half. it is kept in a uniform fluid of density 3p with a hinge at the bottom such that it can freely perform oscillations about horizontal axis through the hinge ( radius of sphere is R) find period of small oscillations? defined & explained in the simplest way possible. Besides giving the explanation of
A solid sphere has density p in the top half and 2p in the bottom half. it is kept in a uniform fluid of density 3p with a hinge at the bottom such that it can freely perform oscillations about horizontal axis through the hinge ( radius of sphere is R) find period of small oscillations?, a detailed solution for A solid sphere has density p in the top half and 2p in the bottom half. it is kept in a uniform fluid of density 3p with a hinge at the bottom such that it can freely perform oscillations about horizontal axis through the hinge ( radius of sphere is R) find period of small oscillations? has been provided alongside types of A solid sphere has density p in the top half and 2p in the bottom half. it is kept in a uniform fluid of density 3p with a hinge at the bottom such that it can freely perform oscillations about horizontal axis through the hinge ( radius of sphere is R) find period of small oscillations? theory, EduRev gives you an
ample number of questions to practice A solid sphere has density p in the top half and 2p in the bottom half. it is kept in a uniform fluid of density 3p with a hinge at the bottom such that it can freely perform oscillations about horizontal axis through the hinge ( radius of sphere is R) find period of small oscillations? tests, examples and also practice UPSC tests.
|
|
Explore Courses for UPSC exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup