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A non-isotropic solid metal cube has coefficients of linear expansion as 5 × 10–5/°C along the x-axis and 5 × 10–6/°C along y-axis and z-axis. If coefficient of volume expansion of the solid is C × 10–6/°C then the value of C is
Correct answer is '60'. Can you explain this answer?
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A non-isotropic solid metal cube has coefficients of linear expansion ...
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A non-isotropic solid metal cube has coefficients of linear expansion ...
Let's call the coefficients of linear expansion for the three dimensions of the cube as αx, αy, and αz, respectively.
According to the problem statement, αx = 5αy = 10αz.
We know that the thermal expansion of a material can be expressed as ΔL = αΔT L0, where ΔL is the change in length, α is the coefficient of linear expansion, ΔT is the change in temperature, and L0 is the original length.
If we subject the cube to a temperature change ΔT, then the change in length in the x-direction would be ΔLx = αxΔT L0x, where L0x is the original length in the x-direction. Similarly, the change in length in the y-direction would be ΔLy = αyΔT L0y, and the change in length in the z-direction would be ΔLz = αzΔT L0z.
Since the cube is non-isotropic, its original lengths L0x, L0y, and L0z are not necessarily equal. Let's assume that the original length of the cube in the x-direction is Lx, in the y-direction is Ly, and in the z-direction is Lz.
Then, the change in volume of the cube can be calculated as follows:
ΔV = (Lx + ΔLx)(Ly + ΔLy)(Lz + ΔLz) - LxLyLz
Expanding this equation and keeping only the first-order terms in ΔLx, ΔLy, and ΔLz, we get:
ΔV ≈ (LxLyLz)(αxΔT + αyΔT + αzΔT)
Substituting the values of αx, αy, and αz from the problem statement, we get:
ΔV ≈ (LxLyLz)(5αΔT + αΔT + 10αΔT)
ΔV ≈ (LxLyLz)(16αΔT)
Therefore, the change in volume of the cube is proportional to the original volume of the cube and the coefficient of linear expansion, and is given by ΔV = (LxLyLz)(16αΔT).
According to the problem statement, αx = 5αy = 10αz.
We know that the thermal expansion of a material can be expressed as ΔL = αΔT L0, where ΔL is the change in length, α is the coefficient of linear expansion, ΔT is the change in temperature, and L0 is the original length.
If we subject the cube to a temperature change ΔT, then the change in length in the x-direction would be ΔLx = αxΔT L0x, where L0x is the original length in the x-direction. Similarly, the change in length in the y-direction would be ΔLy = αyΔT L0y, and the change in length in the z-direction would be ΔLz = αzΔT L0z.
Since the cube is non-isotropic, its original lengths L0x, L0y, and L0z are not necessarily equal. Let's assume that the original length of the cube in the x-direction is Lx, in the y-direction is Ly, and in the z-direction is Lz.
Then, the change in volume of the cube can be calculated as follows:
ΔV = (Lx + ΔLx)(Ly + ΔLy)(Lz + ΔLz) - LxLyLz
Expanding this equation and keeping only the first-order terms in ΔLx, ΔLy, and ΔLz, we get:
ΔV ≈ (LxLyLz)(αxΔT + αyΔT + αzΔT)
Substituting the values of αx, αy, and αz from the problem statement, we get:
ΔV ≈ (LxLyLz)(5αΔT + αΔT + 10αΔT)
ΔV ≈ (LxLyLz)(16αΔT)
Therefore, the change in volume of the cube is proportional to the original volume of the cube and the coefficient of linear expansion, and is given by ΔV = (LxLyLz)(16αΔT).
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A non-isotropic solid metal cube has coefficients of linear expansion ...
Correct answer is '60'
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A non-isotropic solid metal cube has coefficients of linear expansion as 5 × 10–5/°C along the x-axis and 5 × 10–6/°C along y-axis and z-axis. If coefficient of volume expansion of the solid is C × 10–6/°C then the value of C isCorrect answer is '60'. Can you explain this answer?
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A non-isotropic solid metal cube has coefficients of linear expansion as 5 × 10–5/°C along the x-axis and 5 × 10–6/°C along y-axis and z-axis. If coefficient of volume expansion of the solid is C × 10–6/°C then the value of C isCorrect answer is '60'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A non-isotropic solid metal cube has coefficients of linear expansion as 5 × 10–5/°C along the x-axis and 5 × 10–6/°C along y-axis and z-axis. If coefficient of volume expansion of the solid is C × 10–6/°C then the value of C isCorrect answer is '60'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A non-isotropic solid metal cube has coefficients of linear expansion as 5 × 10–5/°C along the x-axis and 5 × 10–6/°C along y-axis and z-axis. If coefficient of volume expansion of the solid is C × 10–6/°C then the value of C isCorrect answer is '60'. Can you explain this answer?.
A non-isotropic solid metal cube has coefficients of linear expansion as 5 × 10–5/°C along the x-axis and 5 × 10–6/°C along y-axis and z-axis. If coefficient of volume expansion of the solid is C × 10–6/°C then the value of C isCorrect answer is '60'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A non-isotropic solid metal cube has coefficients of linear expansion as 5 × 10–5/°C along the x-axis and 5 × 10–6/°C along y-axis and z-axis. If coefficient of volume expansion of the solid is C × 10–6/°C then the value of C isCorrect answer is '60'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A non-isotropic solid metal cube has coefficients of linear expansion as 5 × 10–5/°C along the x-axis and 5 × 10–6/°C along y-axis and z-axis. If coefficient of volume expansion of the solid is C × 10–6/°C then the value of C isCorrect answer is '60'. Can you explain this answer?.
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