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Consider a hollow cylinder of radius R rolling without slipping on an inclined plane. The mass of the cylinder is M and its moment of inertia about its own axis is . The axis of the cylinder is connected to a fixed wall at the top of the inclined plane by a light spring of spring constant k as shown in the figure. In equilibrium, the spring has an extension and system has same kinetic energy in all the situations quoted below at equilibrium. Mark the CORRECT statement(s).?
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Consider a hollow cylinder of radius R rolling without slipping on an ...
Introduction
Consider a hollow cylinder of radius R rolling without slipping on an inclined plane. The axis of the cylinder is connected to a fixed wall at the top of the inclined plane by a light spring of spring constant k as shown in the figure. In equilibrium, the spring has an extension and system has same kinetic energy in all the situations quoted below at equilibrium.
Statement
Mark the CORRECT statement(s).
Solution
There are four situations that need to be analyzed to determine which statement(s) are correct. Let's consider each situation one by one.
Situation 1: The cylinder is at rest on the inclined plane.
In this situation, the cylinder is not moving, so the kinetic energy is zero. The spring is compressed by a distance x, where x is the equilibrium position of the spring. The gravitational force on the cylinder is balanced by the normal force, so the net force on the cylinder is zero. The spring force is also zero, so the spring is only compressed due to the weight of the cylinder. Therefore, the extension of the spring is given by:
kx = Mg sinθ
where k is the spring constant, M is the mass of the cylinder, g is the acceleration due to gravity, and θ is the angle of inclination of the plane.
Situation 2: The cylinder is rolling down the inclined plane without slipping.
In this situation, the cylinder is rolling down the inclined plane without slipping, so the kinetic energy is given by:
K = 1/2 Iw^2 + 1/2 Mv^2
where I is the moment of inertia of the cylinder about its own axis, w is the angular velocity of the cylinder, and v is the linear velocity of the center of mass of the cylinder. The spring is compressed by a distance x, where x is the equilibrium position of the spring. The gravitational force on the cylinder is balanced by the normal force, so the net force on the cylinder is given by:
Mg sinθ - kx = Ma
where a is the acceleration of the center of mass of the cylinder. The torque on the cylinder about its own axis is given by:
Iα = Mg R sinθ - kx R
where α is the angular acceleration of the cylinder. Since the cylinder is rolling without slipping, we have:
α = a/R
Substituting this into the equation for torque, we get:
Ia/R = Mg R sinθ - kx R
Solving for a, we get:
a = (Mg sinθ - kx)/ (M + I/R^2)
Therefore, the extension of the spring is given by:
kx = (M + I/R^2)g sinθ / (1 + I/MR^2)
Situation 3: The cylinder is sliding down the inclined plane without rolling.
In this situation, the cylinder is sliding down the inclined plane without rolling, so the kinetic energy is given by:
K = 1/2 Mv^2
where v is the velocity of the center of mass of the cylinder. The spring is compressed by a distance x, where x is the equilibrium position of the spring. The gravitational force on the cylinder is balanced by the normal force, so the net force on the cylinder is given by:
Mg sinθ - kx
Consider a hollow cylinder of radius R rolling without slipping on an inclined plane. The axis of the cylinder is connected to a fixed wall at the top of the inclined plane by a light spring of spring constant k as shown in the figure. In equilibrium, the spring has an extension and system has same kinetic energy in all the situations quoted below at equilibrium.
Statement
Mark the CORRECT statement(s).
Solution
There are four situations that need to be analyzed to determine which statement(s) are correct. Let's consider each situation one by one.
Situation 1: The cylinder is at rest on the inclined plane.
In this situation, the cylinder is not moving, so the kinetic energy is zero. The spring is compressed by a distance x, where x is the equilibrium position of the spring. The gravitational force on the cylinder is balanced by the normal force, so the net force on the cylinder is zero. The spring force is also zero, so the spring is only compressed due to the weight of the cylinder. Therefore, the extension of the spring is given by:
kx = Mg sinθ
where k is the spring constant, M is the mass of the cylinder, g is the acceleration due to gravity, and θ is the angle of inclination of the plane.
Situation 2: The cylinder is rolling down the inclined plane without slipping.
In this situation, the cylinder is rolling down the inclined plane without slipping, so the kinetic energy is given by:
K = 1/2 Iw^2 + 1/2 Mv^2
where I is the moment of inertia of the cylinder about its own axis, w is the angular velocity of the cylinder, and v is the linear velocity of the center of mass of the cylinder. The spring is compressed by a distance x, where x is the equilibrium position of the spring. The gravitational force on the cylinder is balanced by the normal force, so the net force on the cylinder is given by:
Mg sinθ - kx = Ma
where a is the acceleration of the center of mass of the cylinder. The torque on the cylinder about its own axis is given by:
Iα = Mg R sinθ - kx R
where α is the angular acceleration of the cylinder. Since the cylinder is rolling without slipping, we have:
α = a/R
Substituting this into the equation for torque, we get:
Ia/R = Mg R sinθ - kx R
Solving for a, we get:
a = (Mg sinθ - kx)/ (M + I/R^2)
Therefore, the extension of the spring is given by:
kx = (M + I/R^2)g sinθ / (1 + I/MR^2)
Situation 3: The cylinder is sliding down the inclined plane without rolling.
In this situation, the cylinder is sliding down the inclined plane without rolling, so the kinetic energy is given by:
K = 1/2 Mv^2
where v is the velocity of the center of mass of the cylinder. The spring is compressed by a distance x, where x is the equilibrium position of the spring. The gravitational force on the cylinder is balanced by the normal force, so the net force on the cylinder is given by:
Mg sinθ - kx
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Consider a hollow cylinder of radius R rolling without slipping on an inclined plane. The mass of the cylinder is M and its moment of inertia about its own axis is . The axis of the cylinder is connected to a fixed wall at the top of the inclined plane by a light spring of spring constant k as shown in the figure. In equilibrium, the spring has an extension and system has same kinetic energy in all the situations quoted below at equilibrium. Mark the CORRECT statement(s).?
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Consider a hollow cylinder of radius R rolling without slipping on an inclined plane. The mass of the cylinder is M and its moment of inertia about its own axis is . The axis of the cylinder is connected to a fixed wall at the top of the inclined plane by a light spring of spring constant k as shown in the figure. In equilibrium, the spring has an extension and system has same kinetic energy in all the situations quoted below at equilibrium. Mark the CORRECT statement(s).? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Consider a hollow cylinder of radius R rolling without slipping on an inclined plane. The mass of the cylinder is M and its moment of inertia about its own axis is . The axis of the cylinder is connected to a fixed wall at the top of the inclined plane by a light spring of spring constant k as shown in the figure. In equilibrium, the spring has an extension and system has same kinetic energy in all the situations quoted below at equilibrium. Mark the CORRECT statement(s).? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider a hollow cylinder of radius R rolling without slipping on an inclined plane. The mass of the cylinder is M and its moment of inertia about its own axis is . The axis of the cylinder is connected to a fixed wall at the top of the inclined plane by a light spring of spring constant k as shown in the figure. In equilibrium, the spring has an extension and system has same kinetic energy in all the situations quoted below at equilibrium. Mark the CORRECT statement(s).?.
Consider a hollow cylinder of radius R rolling without slipping on an inclined plane. The mass of the cylinder is M and its moment of inertia about its own axis is . The axis of the cylinder is connected to a fixed wall at the top of the inclined plane by a light spring of spring constant k as shown in the figure. In equilibrium, the spring has an extension and system has same kinetic energy in all the situations quoted below at equilibrium. Mark the CORRECT statement(s).? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Consider a hollow cylinder of radius R rolling without slipping on an inclined plane. The mass of the cylinder is M and its moment of inertia about its own axis is . The axis of the cylinder is connected to a fixed wall at the top of the inclined plane by a light spring of spring constant k as shown in the figure. In equilibrium, the spring has an extension and system has same kinetic energy in all the situations quoted below at equilibrium. Mark the CORRECT statement(s).? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider a hollow cylinder of radius R rolling without slipping on an inclined plane. The mass of the cylinder is M and its moment of inertia about its own axis is . The axis of the cylinder is connected to a fixed wall at the top of the inclined plane by a light spring of spring constant k as shown in the figure. In equilibrium, the spring has an extension and system has same kinetic energy in all the situations quoted below at equilibrium. Mark the CORRECT statement(s).?.
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Consider a hollow cylinder of radius R rolling without slipping on an inclined plane. The mass of the cylinder is M and its moment of inertia about its own axis is . The axis of the cylinder is connected to a fixed wall at the top of the inclined plane by a light spring of spring constant k as shown in the figure. In equilibrium, the spring has an extension and system has same kinetic energy in all the situations quoted below at equilibrium. Mark the CORRECT statement(s).?, a detailed solution for Consider a hollow cylinder of radius R rolling without slipping on an inclined plane. The mass of the cylinder is M and its moment of inertia about its own axis is . The axis of the cylinder is connected to a fixed wall at the top of the inclined plane by a light spring of spring constant k as shown in the figure. In equilibrium, the spring has an extension and system has same kinetic energy in all the situations quoted below at equilibrium. Mark the CORRECT statement(s).? has been provided alongside types of Consider a hollow cylinder of radius R rolling without slipping on an inclined plane. The mass of the cylinder is M and its moment of inertia about its own axis is . The axis of the cylinder is connected to a fixed wall at the top of the inclined plane by a light spring of spring constant k as shown in the figure. In equilibrium, the spring has an extension and system has same kinetic energy in all the situations quoted below at equilibrium. Mark the CORRECT statement(s).? theory, EduRev gives you an
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