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A circular lamina moves with its plane parallel to the x-y plane of a reference frame S at rest. Assuming its motion to be along the axis of x (or y), calculate the velocity at which surface area would appear to be reduced to half to an observer in frame S?
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A circular lamina moves with its plane parallel to the x-y plane of a ...
Solution:
**Introduction**
In this problem, we are given a circular lamina moving with its plane parallel to the x-y plane of a reference frame S at rest. We need to find the velocity at which surface area would appear to be reduced to half to an observer in frame S.
**Explanation**
Let us assume that the circular lamina is moving along the x-axis with a velocity v. The lamina has a radius R and its center is at the origin of the reference frame S.
The surface area of the circular lamina is given by:
A = πR^2
Let us consider a small element of area dA on the circular lamina. This element of area is moving with a velocity v along the x-axis.
The apparent area of the element of area dA as observed by an observer in frame S is given by:
dA' = dA/γ
where γ is the Lorentz factor given by:
γ = 1/√(1-v^2/c^2)
where c is the speed of light.
The total surface area of the circular lamina as observed by an observer in frame S is given by:
A' = A/γ
If we want the surface area to be reduced to half, we need to find the velocity v at which A' = A/2.
So, we have:
A/γ = A/2
or
γ = 2
Substituting the value of γ in the equation for Lorentz factor, we get:
2 = 1/√(1-v^2/c^2)
Squaring both sides, we get:
4 = 1/(1-v^2/c^2)
or
v^2/c^2 = 3/4
or
v = c/2√3 ≈ 0.2887c
Therefore, the velocity at which the surface area of the circular lamina would appear to be reduced to half to an observer in frame S is approximately 0.2887 times the speed of light.
**Conclusion**
In this problem, we have calculated the velocity at which the surface area of a circular lamina moving along the x-axis would appear to be reduced to half to an observer in frame S. The velocity is approximately 0.2887 times the speed of light.
**Introduction**
In this problem, we are given a circular lamina moving with its plane parallel to the x-y plane of a reference frame S at rest. We need to find the velocity at which surface area would appear to be reduced to half to an observer in frame S.
**Explanation**
Let us assume that the circular lamina is moving along the x-axis with a velocity v. The lamina has a radius R and its center is at the origin of the reference frame S.
The surface area of the circular lamina is given by:
A = πR^2
Let us consider a small element of area dA on the circular lamina. This element of area is moving with a velocity v along the x-axis.
The apparent area of the element of area dA as observed by an observer in frame S is given by:
dA' = dA/γ
where γ is the Lorentz factor given by:
γ = 1/√(1-v^2/c^2)
where c is the speed of light.
The total surface area of the circular lamina as observed by an observer in frame S is given by:
A' = A/γ
If we want the surface area to be reduced to half, we need to find the velocity v at which A' = A/2.
So, we have:
A/γ = A/2
or
γ = 2
Substituting the value of γ in the equation for Lorentz factor, we get:
2 = 1/√(1-v^2/c^2)
Squaring both sides, we get:
4 = 1/(1-v^2/c^2)
or
v^2/c^2 = 3/4
or
v = c/2√3 ≈ 0.2887c
Therefore, the velocity at which the surface area of the circular lamina would appear to be reduced to half to an observer in frame S is approximately 0.2887 times the speed of light.
**Conclusion**
In this problem, we have calculated the velocity at which the surface area of a circular lamina moving along the x-axis would appear to be reduced to half to an observer in frame S. The velocity is approximately 0.2887 times the speed of light.
Community Answer
A circular lamina moves with its plane parallel to the x-y plane of a ...
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A circular lamina moves with its plane parallel to the x-y plane of a reference frame S at rest. Assuming its motion to be along the axis of x (or y), calculate the velocity at which surface area would appear to be reduced to half to an observer in frame S?
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A circular lamina moves with its plane parallel to the x-y plane of a reference frame S at rest. Assuming its motion to be along the axis of x (or y), calculate the velocity at which surface area would appear to be reduced to half to an observer in frame S? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about A circular lamina moves with its plane parallel to the x-y plane of a reference frame S at rest. Assuming its motion to be along the axis of x (or y), calculate the velocity at which surface area would appear to be reduced to half to an observer in frame S? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A circular lamina moves with its plane parallel to the x-y plane of a reference frame S at rest. Assuming its motion to be along the axis of x (or y), calculate the velocity at which surface area would appear to be reduced to half to an observer in frame S?.
A circular lamina moves with its plane parallel to the x-y plane of a reference frame S at rest. Assuming its motion to be along the axis of x (or y), calculate the velocity at which surface area would appear to be reduced to half to an observer in frame S? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about A circular lamina moves with its plane parallel to the x-y plane of a reference frame S at rest. Assuming its motion to be along the axis of x (or y), calculate the velocity at which surface area would appear to be reduced to half to an observer in frame S? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A circular lamina moves with its plane parallel to the x-y plane of a reference frame S at rest. Assuming its motion to be along the axis of x (or y), calculate the velocity at which surface area would appear to be reduced to half to an observer in frame S?.
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