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In the context of small oscillations, which one of the following does NOT apply to the normal coordinates?
- a)Each normal coordinate has an eigen-frequency associated with it
- b)The normal coordinates are orthogonal to one another
- c)The normal coordinates are all independent
- d)The potential energy of the system is a sum of squares of the normal coordinates with constant coefficients
Correct answer is option 'B'. Can you explain this answer?
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In the context of small oscillations, which one of the following does ...
We begin with the one-dimensional case of a particle oscillating about a local minimum of the potential energy We'll assume that near the minimum, call it , the potential is well described by the leading second-order term, , so we're taking the zero of potential at assuming that the second derivative , and (for now) neglecting higher order terms.
To simplify the equations, we'll also move theorigin to , so
replacing the second derivative with the standard "spring constant" expression.
This equation has solution
(This can, of course, also be derived from the Lagrangian, easily shown to be )
The physical motion corresponding to the amplitudes eigenvector has two constants of integration (amplitude and phase), often written in terms of a single complex number, that is,
with real.
Clearly, this is the mode in which the two pendulums are in sync, oscillating at their natural frequency, with the spring playing no role.
In physics, this mathematical eigenstate of the matrix is called a normal mode of oscillation. In a normal mode, all parts of the system oscillate at a single frequency, given by the eigenvalue.
The other normal mode
where we have written . Here the system is oscillating with the single frequency , the pendulums are now exactly out of phase, so when they part the spring pulls them back to the center, thereby increasing the system oscillation frequency
The matrix structure can be clarified by separating out the spring contribution:
Most Upvoted Answer
In the context of small oscillations, which one of the following does ...
Introduction:
In the context of small oscillations, normal coordinates are used to describe the motion of a system with multiple degrees of freedom. These coordinates are linear combinations of the original coordinates and allow for a simpler analysis of the system's behavior. It is important to understand the properties of normal coordinates in order to correctly analyze the system.
Explanation:
Among the given options, option 'B' does not apply to the normal coordinates. Let's discuss each option in detail:
a) Each normal coordinate has an eigen-frequency associated with it:
Normal coordinates are defined in such a way that each coordinate oscillates with a specific frequency, known as the eigen-frequency. These eigen-frequencies are characteristic of the system and depend on the masses, spring constants, and other parameters involved. Therefore, option 'A' is correct.
b) The normal coordinates are orthogonal to one another:
Orthogonal means that two vectors are perpendicular to each other. In the context of normal coordinates, this property does not hold. Normal coordinates are defined as linear combinations of the original coordinates, and the coefficients in these combinations are generally not chosen to ensure orthogonality. Therefore, option 'B' is incorrect.
c) The normal coordinates are all independent:
Independence means that the motion described by one normal coordinate does not affect the motion described by another normal coordinate. In the context of small oscillations, this property holds true for normal coordinates. The motion along each normal coordinate is independent of the motion along other normal coordinates. Therefore, option 'C' is correct.
d) The potential energy of the system is a sum of squares of the normal coordinates with constant coefficients:
In the context of small oscillations, the potential energy of the system can be expressed as a sum of squares of the normal coordinates multiplied by constant coefficients. This expression arises from the fact that the potential energy can be expanded in terms of the normal coordinates using Taylor series approximation. Therefore, option 'D' is correct.
Conclusion:
In the context of small oscillations, the normal coordinates have eigen-frequencies associated with them, are independent of each other, and the potential energy of the system can be expressed as a sum of squares of the normal coordinates with constant coefficients. However, normal coordinates are not orthogonal to each other.
In the context of small oscillations, normal coordinates are used to describe the motion of a system with multiple degrees of freedom. These coordinates are linear combinations of the original coordinates and allow for a simpler analysis of the system's behavior. It is important to understand the properties of normal coordinates in order to correctly analyze the system.
Explanation:
Among the given options, option 'B' does not apply to the normal coordinates. Let's discuss each option in detail:
a) Each normal coordinate has an eigen-frequency associated with it:
Normal coordinates are defined in such a way that each coordinate oscillates with a specific frequency, known as the eigen-frequency. These eigen-frequencies are characteristic of the system and depend on the masses, spring constants, and other parameters involved. Therefore, option 'A' is correct.
b) The normal coordinates are orthogonal to one another:
Orthogonal means that two vectors are perpendicular to each other. In the context of normal coordinates, this property does not hold. Normal coordinates are defined as linear combinations of the original coordinates, and the coefficients in these combinations are generally not chosen to ensure orthogonality. Therefore, option 'B' is incorrect.
c) The normal coordinates are all independent:
Independence means that the motion described by one normal coordinate does not affect the motion described by another normal coordinate. In the context of small oscillations, this property holds true for normal coordinates. The motion along each normal coordinate is independent of the motion along other normal coordinates. Therefore, option 'C' is correct.
d) The potential energy of the system is a sum of squares of the normal coordinates with constant coefficients:
In the context of small oscillations, the potential energy of the system can be expressed as a sum of squares of the normal coordinates multiplied by constant coefficients. This expression arises from the fact that the potential energy can be expanded in terms of the normal coordinates using Taylor series approximation. Therefore, option 'D' is correct.
Conclusion:
In the context of small oscillations, the normal coordinates have eigen-frequencies associated with them, are independent of each other, and the potential energy of the system can be expressed as a sum of squares of the normal coordinates with constant coefficients. However, normal coordinates are not orthogonal to each other.
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In the context of small oscillations, which one of the following does NOT apply to the normal coordinates?a)Each normal coordinate has an eigen-frequency associated with itb)The normal coordinates are orthogonal to one anotherc)The normal coordinates are all independentd)The potential energy of the system is a sum of squares of the normal coordinates with constant coefficientsCorrect answer is option 'B'. Can you explain this answer?
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In the context of small oscillations, which one of the following does NOT apply to the normal coordinates?a)Each normal coordinate has an eigen-frequency associated with itb)The normal coordinates are orthogonal to one anotherc)The normal coordinates are all independentd)The potential energy of the system is a sum of squares of the normal coordinates with constant coefficientsCorrect answer is option 'B'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about In the context of small oscillations, which one of the following does NOT apply to the normal coordinates?a)Each normal coordinate has an eigen-frequency associated with itb)The normal coordinates are orthogonal to one anotherc)The normal coordinates are all independentd)The potential energy of the system is a sum of squares of the normal coordinates with constant coefficientsCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the context of small oscillations, which one of the following does NOT apply to the normal coordinates?a)Each normal coordinate has an eigen-frequency associated with itb)The normal coordinates are orthogonal to one anotherc)The normal coordinates are all independentd)The potential energy of the system is a sum of squares of the normal coordinates with constant coefficientsCorrect answer is option 'B'. Can you explain this answer?.
In the context of small oscillations, which one of the following does NOT apply to the normal coordinates?a)Each normal coordinate has an eigen-frequency associated with itb)The normal coordinates are orthogonal to one anotherc)The normal coordinates are all independentd)The potential energy of the system is a sum of squares of the normal coordinates with constant coefficientsCorrect answer is option 'B'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about In the context of small oscillations, which one of the following does NOT apply to the normal coordinates?a)Each normal coordinate has an eigen-frequency associated with itb)The normal coordinates are orthogonal to one anotherc)The normal coordinates are all independentd)The potential energy of the system is a sum of squares of the normal coordinates with constant coefficientsCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the context of small oscillations, which one of the following does NOT apply to the normal coordinates?a)Each normal coordinate has an eigen-frequency associated with itb)The normal coordinates are orthogonal to one anotherc)The normal coordinates are all independentd)The potential energy of the system is a sum of squares of the normal coordinates with constant coefficientsCorrect answer is option 'B'. Can you explain this answer?.
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