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Two loops P and Q are made from a uniform wire. The radii of P and Q are R1 and R2 respectively and their moment of inertia about axes normally through center are I1 and I2 respectively. If I2/I1 = 4 then find R2/R1?
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Two loops P and Q are made from a uniform wire. The radii of P and Q a...
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Two loops P and Q are made from a uniform wire. The radii of P and Q a...
Given:
- Loops P and Q are made from a uniform wire.
- The radii of P and Q are R1 and R2 respectively.
- The moment of inertia about axes normally through the center of P and Q are I1 and I2 respectively.
- I2/I1 = 4
To find:
The ratio of R2 to R1 (R2/R1).
Solution:
We know that the moment of inertia (I) of a circular loop is given by the formula:
I = (1/2) * m * r^2
where m is the mass of the loop and r is the radius.
Step 1: Deriving the Ratio of Moments of Inertia
Since the wire used to make the loops is uniform, the mass per unit length is the same for both loops P and Q. Let's assume it is represented by λ.
The mass of each loop can be calculated by multiplying the mass per unit length by the circumference of the loop:
m1 = λ * 2πR1
m2 = λ * 2πR2
Substituting these values into the formula for moment of inertia, we get:
I1 = (1/2) * (λ * 2πR1) * R1^2
I2 = (1/2) * (λ * 2πR2) * R2^2
Simplifying these equations, we get:
I1 = λ * π * R1^3
I2 = λ * π * R2^3
Now, we can find the ratio of the moments of inertia:
I2/I1 = (λ * π * R2^3) / (λ * π * R1^3)
I2/I1 = R2^3 / R1^3
Given that I2/I1 = 4, we can equate the above equation to 4:
4 = R2^3 / R1^3
Step 2: Finding R2/R1
To find the ratio of R2 to R1, we take the cube root of both sides of the equation:
∛4 = ∛(R2^3 / R1^3)
∛4 = R2 / R1
Therefore, R2/R1 = ∛4
Final Answer:
The ratio of R2 to R1 is ∛4, which is approximately 1.587.
- Loops P and Q are made from a uniform wire.
- The radii of P and Q are R1 and R2 respectively.
- The moment of inertia about axes normally through the center of P and Q are I1 and I2 respectively.
- I2/I1 = 4
To find:
The ratio of R2 to R1 (R2/R1).
Solution:
We know that the moment of inertia (I) of a circular loop is given by the formula:
I = (1/2) * m * r^2
where m is the mass of the loop and r is the radius.
Step 1: Deriving the Ratio of Moments of Inertia
Since the wire used to make the loops is uniform, the mass per unit length is the same for both loops P and Q. Let's assume it is represented by λ.
The mass of each loop can be calculated by multiplying the mass per unit length by the circumference of the loop:
m1 = λ * 2πR1
m2 = λ * 2πR2
Substituting these values into the formula for moment of inertia, we get:
I1 = (1/2) * (λ * 2πR1) * R1^2
I2 = (1/2) * (λ * 2πR2) * R2^2
Simplifying these equations, we get:
I1 = λ * π * R1^3
I2 = λ * π * R2^3
Now, we can find the ratio of the moments of inertia:
I2/I1 = (λ * π * R2^3) / (λ * π * R1^3)
I2/I1 = R2^3 / R1^3
Given that I2/I1 = 4, we can equate the above equation to 4:
4 = R2^3 / R1^3
Step 2: Finding R2/R1
To find the ratio of R2 to R1, we take the cube root of both sides of the equation:
∛4 = ∛(R2^3 / R1^3)
∛4 = R2 / R1
Therefore, R2/R1 = ∛4
Final Answer:
The ratio of R2 to R1 is ∛4, which is approximately 1.587.
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Two loops P and Q are made from a uniform wire. The radii of P and Q are R1 and R2 respectively and their moment of inertia about axes normally through center are I1 and I2 respectively. If I2/I1 = 4 then find R2/R1?
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Two loops P and Q are made from a uniform wire. The radii of P and Q are R1 and R2 respectively and their moment of inertia about axes normally through center are I1 and I2 respectively. If I2/I1 = 4 then find R2/R1? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about Two loops P and Q are made from a uniform wire. The radii of P and Q are R1 and R2 respectively and their moment of inertia about axes normally through center are I1 and I2 respectively. If I2/I1 = 4 then find R2/R1? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two loops P and Q are made from a uniform wire. The radii of P and Q are R1 and R2 respectively and their moment of inertia about axes normally through center are I1 and I2 respectively. If I2/I1 = 4 then find R2/R1?.
Two loops P and Q are made from a uniform wire. The radii of P and Q are R1 and R2 respectively and their moment of inertia about axes normally through center are I1 and I2 respectively. If I2/I1 = 4 then find R2/R1? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about Two loops P and Q are made from a uniform wire. The radii of P and Q are R1 and R2 respectively and their moment of inertia about axes normally through center are I1 and I2 respectively. If I2/I1 = 4 then find R2/R1? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two loops P and Q are made from a uniform wire. The radii of P and Q are R1 and R2 respectively and their moment of inertia about axes normally through center are I1 and I2 respectively. If I2/I1 = 4 then find R2/R1?.
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