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A ring of mass m and radius r has four particles each of mass m attached to the ring the centre of ring has the speed v.the kinetic energy of the system ?
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A ring of mass m and radius r has four particles each of mass m attach...
The kinetic energy of the system of particles attached to the ring can be determined by considering the kinetic energy of each individual particle and the kinetic energy of the ring itself. Let's break down the calculation of the total kinetic energy step by step:
1. Kinetic Energy of the Particles:
Each particle attached to the ring has mass m and is moving with the same speed v as the center of the ring. The kinetic energy of each particle can be calculated using the formula: KE = (1/2)mv^2. Since there are four particles, the total kinetic energy of the particles is 4 times the kinetic energy of an individual particle.
2. Kinetic Energy of the Ring:
The ring also contributes to the total kinetic energy of the system. The ring has a mass of m and is rotating about its center with the same speed v. The kinetic energy of the ring can be calculated using the formula for rotational kinetic energy: KE = (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity.
The moment of inertia of a ring rotating about its center is given by the formula: I = (1/2)mr^2, where m is the mass of the ring and r is its radius. The angular velocity ω can be determined using the relation ω = v/r, where v is the linear velocity of the center of the ring and r is its radius.
3. Total Kinetic Energy:
To find the total kinetic energy of the system, we need to sum up the kinetic energies of the particles and the ring. Therefore, we can write the equation as follows:
Total KE = (4 * (1/2)mv^2) + ((1/2)Iω^2)
Substituting the values for I and ω, we get:
Total KE = (4 * (1/2)mv^2) + ((1/2)((1/2)mr^2)(v/r)^2)
Simplifying the equation further:
Total KE = 2mv^2 + (1/2)(1/4)mv^2
Total KE = 2mv^2 + (1/8)mv^2
Total KE = (16/8)mv^2 + (1/8)mv^2
Total KE = (17/8)mv^2
Therefore, the kinetic energy of the system of particles attached to the ring is equal to (17/8)mv^2.
1. Kinetic Energy of the Particles:
Each particle attached to the ring has mass m and is moving with the same speed v as the center of the ring. The kinetic energy of each particle can be calculated using the formula: KE = (1/2)mv^2. Since there are four particles, the total kinetic energy of the particles is 4 times the kinetic energy of an individual particle.
2. Kinetic Energy of the Ring:
The ring also contributes to the total kinetic energy of the system. The ring has a mass of m and is rotating about its center with the same speed v. The kinetic energy of the ring can be calculated using the formula for rotational kinetic energy: KE = (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity.
The moment of inertia of a ring rotating about its center is given by the formula: I = (1/2)mr^2, where m is the mass of the ring and r is its radius. The angular velocity ω can be determined using the relation ω = v/r, where v is the linear velocity of the center of the ring and r is its radius.
3. Total Kinetic Energy:
To find the total kinetic energy of the system, we need to sum up the kinetic energies of the particles and the ring. Therefore, we can write the equation as follows:
Total KE = (4 * (1/2)mv^2) + ((1/2)Iω^2)
Substituting the values for I and ω, we get:
Total KE = (4 * (1/2)mv^2) + ((1/2)((1/2)mr^2)(v/r)^2)
Simplifying the equation further:
Total KE = 2mv^2 + (1/2)(1/4)mv^2
Total KE = 2mv^2 + (1/8)mv^2
Total KE = (16/8)mv^2 + (1/8)mv^2
Total KE = (17/8)mv^2
Therefore, the kinetic energy of the system of particles attached to the ring is equal to (17/8)mv^2.
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A ring of mass m and radius r has four particles each of mass m attach...
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A ring of mass m and radius r has four particles each of mass m attached to the ring the centre of ring has the speed v.the kinetic energy of the system ?
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A ring of mass m and radius r has four particles each of mass m attached to the ring the centre of ring has the speed v.the kinetic energy of the system ? 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 A ring of mass m and radius r has four particles each of mass m attached to the ring the centre of ring has the speed v.the kinetic energy of the system ? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A ring of mass m and radius r has four particles each of mass m attached to the ring the centre of ring has the speed v.the kinetic energy of the system ?.
A ring of mass m and radius r has four particles each of mass m attached to the ring the centre of ring has the speed v.the kinetic energy of the system ? 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 A ring of mass m and radius r has four particles each of mass m attached to the ring the centre of ring has the speed v.the kinetic energy of the system ? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A ring of mass m and radius r has four particles each of mass m attached to the ring the centre of ring has the speed v.the kinetic energy of the system ?.
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