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A metallic thin wire has uniform thickness. from this wire, two circular loops of radius r and nr are made. if moment of inertia of second loop about its actual axis is 8 times moment of inertia of first loop about its natural axis. the value of n is. 1)8. 2)4. 3)2. 4)2√2.?
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A metallic thin wire has uniform thickness. from this wire, two circul...
The problem states that a metallic thin wire is used to create two circular loops, one with a radius of r and the other with a radius of nr. We need to find the value of n, given that the moment of inertia of the second loop about its actual axis is 8 times the moment of inertia of the first loop about its natural axis.
Let's break down the problem into smaller parts to find the solution.
**Understanding Moment of Inertia**
- Moment of inertia is a measure of an object's resistance to changes in rotational motion.
- For a circular loop, the moment of inertia about its natural axis passing through the center of the loop is given by the formula: I = 0.5 * m * r^2, where m is the mass of the loop and r is the radius.
**Comparing Moments of Inertia**
- The moment of inertia of the second loop about its actual axis depends on its shape and radius.
- The moment of inertia of the first loop about its natural axis is given by I1 = 0.5 * m1 * r^2.
- The moment of inertia of the second loop about its actual axis is given by I2 = 0.5 * m2 * (nr)^2 = 0.5 * m2 * n^2 * r^2.
**Setting up the Equation**
- According to the problem, I2 = 8 * I1.
- Substituting the moment of inertia formulas, we get 0.5 * m2 * n^2 * r^2 = 8 * 0.5 * m1 * r^2.
**Simplifying the Equation**
- Cancelling common terms, we have m2 * n^2 = 8 * m1.
- Since the wire has uniform thickness, the mass of the wire is directly proportional to its length.
- The length of the wire used to create the first loop is proportional to its circumference, which is 2πr.
- The length of the wire used to create the second loop is proportional to its circumference, which is 2π(nr).
- Therefore, m2/m1 = (2π(nr))/(2πr) = n.
**Solving for n**
- Substituting this value in the equation m2 * n^2 = 8 * m1, we get n * n = 8.
- Taking the square root of both sides, we find n = √8 = 2√2.
Therefore, the value of n is 2√2, which corresponds to option 4.
Let's break down the problem into smaller parts to find the solution.
**Understanding Moment of Inertia**
- Moment of inertia is a measure of an object's resistance to changes in rotational motion.
- For a circular loop, the moment of inertia about its natural axis passing through the center of the loop is given by the formula: I = 0.5 * m * r^2, where m is the mass of the loop and r is the radius.
**Comparing Moments of Inertia**
- The moment of inertia of the second loop about its actual axis depends on its shape and radius.
- The moment of inertia of the first loop about its natural axis is given by I1 = 0.5 * m1 * r^2.
- The moment of inertia of the second loop about its actual axis is given by I2 = 0.5 * m2 * (nr)^2 = 0.5 * m2 * n^2 * r^2.
**Setting up the Equation**
- According to the problem, I2 = 8 * I1.
- Substituting the moment of inertia formulas, we get 0.5 * m2 * n^2 * r^2 = 8 * 0.5 * m1 * r^2.
**Simplifying the Equation**
- Cancelling common terms, we have m2 * n^2 = 8 * m1.
- Since the wire has uniform thickness, the mass of the wire is directly proportional to its length.
- The length of the wire used to create the first loop is proportional to its circumference, which is 2πr.
- The length of the wire used to create the second loop is proportional to its circumference, which is 2π(nr).
- Therefore, m2/m1 = (2π(nr))/(2πr) = n.
**Solving for n**
- Substituting this value in the equation m2 * n^2 = 8 * m1, we get n * n = 8.
- Taking the square root of both sides, we find n = √8 = 2√2.
Therefore, the value of n is 2√2, which corresponds to option 4.
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A metallic thin wire has uniform thickness. from this wire, two circular loops of radius r and nr are made. if moment of inertia of second loop about its actual axis is 8 times moment of inertia of first loop about its natural axis. the value of n is. 1)8. 2)4. 3)2. 4)2√2.?
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A metallic thin wire has uniform thickness. from this wire, two circular loops of radius r and nr are made. if moment of inertia of second loop about its actual axis is 8 times moment of inertia of first loop about its natural axis. the value of n is. 1)8. 2)4. 3)2. 4)2√2.? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about A metallic thin wire has uniform thickness. from this wire, two circular loops of radius r and nr are made. if moment of inertia of second loop about its actual axis is 8 times moment of inertia of first loop about its natural axis. the value of n is. 1)8. 2)4. 3)2. 4)2√2.? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A metallic thin wire has uniform thickness. from this wire, two circular loops of radius r and nr are made. if moment of inertia of second loop about its actual axis is 8 times moment of inertia of first loop about its natural axis. the value of n is. 1)8. 2)4. 3)2. 4)2√2.?.
A metallic thin wire has uniform thickness. from this wire, two circular loops of radius r and nr are made. if moment of inertia of second loop about its actual axis is 8 times moment of inertia of first loop about its natural axis. the value of n is. 1)8. 2)4. 3)2. 4)2√2.? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about A metallic thin wire has uniform thickness. from this wire, two circular loops of radius r and nr are made. if moment of inertia of second loop about its actual axis is 8 times moment of inertia of first loop about its natural axis. the value of n is. 1)8. 2)4. 3)2. 4)2√2.? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A metallic thin wire has uniform thickness. from this wire, two circular loops of radius r and nr are made. if moment of inertia of second loop about its actual axis is 8 times moment of inertia of first loop about its natural axis. the value of n is. 1)8. 2)4. 3)2. 4)2√2.?.
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