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If universal gravitational constant G, speed of light c and mass taken as fundamental P.Q. then dimension of area = G M C then value of , and are – (1) –2, –2, 4 (2) 3, 3, –2 (3) 2, 2, – 4 (4) –3, –3, 2?
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If universal gravitational constant G, speed of light c and mass taken...
To determine the dimensions of area in terms of the universal gravitational constant (G), the speed of light (c), and mass (M), we can use the method of dimensional analysis.
**Step 1: Analyzing the given variables:**
- G: Universal gravitational constant
- c: Speed of light
- M: Mass
**Step 2: Determining the dimensions of the given variables:**
- G has units of N(m/kg)^2. This can be determined from the formula for gravitational force: F = G(Mm/r^2), where F is the force, M and m are masses, and r is the distance between them. Rearranging the formula, we get G = Fr^2/(Mm), where Fr^2 is the force dimension and Mm is the mass dimension. Therefore, the dimensions of G are (force * distance^2) / (mass * mass), which can be simplified to (kg * m^2) / (kg^2) = m^3 / (kg * s^2).
- c has units of m/s. This is the speed of light, which is a fundamental constant with dimensions of distance / time.
- M has units of kg. This is the mass, which is a fundamental quantity with dimensions of mass.
**Step 3: Determining the dimensions of area:**
Let's assume the dimensions of area are A^α M^β C^γ, where A represents area.
From Step 2, we know that the dimensions of G are m^3 / (kg * s^2), the dimensions of c are m/s, and the dimensions of M are kg.
The dimensions of area can be determined by substituting these dimensions into the given equation:
G^α M^β C^γ = (m^3 / (kg * s^2))^α (kg)^β (m/s)^γ.
**Step 4: Equating the dimensions on both sides of the equation:**
On the left side of the equation, the dimensions are (m^3 / (kg * s^2))^α (kg)^β (m/s)^γ.
On the right side of the equation, the dimensions are m^(3α + γ) / (kg^(α + β) * s^(2α)).
Equating the dimensions on both sides, we get:
3α + γ = 1 (for the dimensions of length)
α + β = -1 (for the dimensions of mass)
2α = 0 (for the dimensions of time)
Solving these equations, we find:
α = 0
β = -1
γ = 1
Therefore, the dimensions of area (A) in terms of G, c, and M are A^0 M^-1 C^1, which simplifies to C. So, the correct answer is (4) –3, –3, 2, where α = -3, β = -3, and γ = 2.
**Step 1: Analyzing the given variables:**
- G: Universal gravitational constant
- c: Speed of light
- M: Mass
**Step 2: Determining the dimensions of the given variables:**
- G has units of N(m/kg)^2. This can be determined from the formula for gravitational force: F = G(Mm/r^2), where F is the force, M and m are masses, and r is the distance between them. Rearranging the formula, we get G = Fr^2/(Mm), where Fr^2 is the force dimension and Mm is the mass dimension. Therefore, the dimensions of G are (force * distance^2) / (mass * mass), which can be simplified to (kg * m^2) / (kg^2) = m^3 / (kg * s^2).
- c has units of m/s. This is the speed of light, which is a fundamental constant with dimensions of distance / time.
- M has units of kg. This is the mass, which is a fundamental quantity with dimensions of mass.
**Step 3: Determining the dimensions of area:**
Let's assume the dimensions of area are A^α M^β C^γ, where A represents area.
From Step 2, we know that the dimensions of G are m^3 / (kg * s^2), the dimensions of c are m/s, and the dimensions of M are kg.
The dimensions of area can be determined by substituting these dimensions into the given equation:
G^α M^β C^γ = (m^3 / (kg * s^2))^α (kg)^β (m/s)^γ.
**Step 4: Equating the dimensions on both sides of the equation:**
On the left side of the equation, the dimensions are (m^3 / (kg * s^2))^α (kg)^β (m/s)^γ.
On the right side of the equation, the dimensions are m^(3α + γ) / (kg^(α + β) * s^(2α)).
Equating the dimensions on both sides, we get:
3α + γ = 1 (for the dimensions of length)
α + β = -1 (for the dimensions of mass)
2α = 0 (for the dimensions of time)
Solving these equations, we find:
α = 0
β = -1
γ = 1
Therefore, the dimensions of area (A) in terms of G, c, and M are A^0 M^-1 C^1, which simplifies to C. So, the correct answer is (4) –3, –3, 2, where α = -3, β = -3, and γ = 2.
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If universal gravitational constant G, speed of light c and mass taken as fundamental P.Q. then dimension of area = G M C then value of , and are – (1) –2, –2, 4 (2) 3, 3, –2 (3) 2, 2, – 4 (4) –3, –3, 2?
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If universal gravitational constant G, speed of light c and mass taken as fundamental P.Q. then dimension of area = G M C then value of , and are – (1) –2, –2, 4 (2) 3, 3, –2 (3) 2, 2, – 4 (4) –3, –3, 2? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about If universal gravitational constant G, speed of light c and mass taken as fundamental P.Q. then dimension of area = G M C then value of , and are – (1) –2, –2, 4 (2) 3, 3, –2 (3) 2, 2, – 4 (4) –3, –3, 2? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If universal gravitational constant G, speed of light c and mass taken as fundamental P.Q. then dimension of area = G M C then value of , and are – (1) –2, –2, 4 (2) 3, 3, –2 (3) 2, 2, – 4 (4) –3, –3, 2?.
If universal gravitational constant G, speed of light c and mass taken as fundamental P.Q. then dimension of area = G M C then value of , and are – (1) –2, –2, 4 (2) 3, 3, –2 (3) 2, 2, – 4 (4) –3, –3, 2? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about If universal gravitational constant G, speed of light c and mass taken as fundamental P.Q. then dimension of area = G M C then value of , and are – (1) –2, –2, 4 (2) 3, 3, –2 (3) 2, 2, – 4 (4) –3, –3, 2? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If universal gravitational constant G, speed of light c and mass taken as fundamental P.Q. then dimension of area = G M C then value of , and are – (1) –2, –2, 4 (2) 3, 3, –2 (3) 2, 2, – 4 (4) –3, –3, 2?.
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