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An inclined plane makes an angle of 30° with the horizontal. A solid sphere rolling down on this inclined plane from rest without slipping, has a linear acceleration, equal to?
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An inclined plane makes an angle of 30° with the horizontal. A solid s...
Understanding the Problem
When a solid sphere rolls down an inclined plane, the forces acting on it include gravitational force, normal force, and friction. The key to finding the linear acceleration is to analyze these forces and the moment of inertia of the sphere.
Forces on the Sphere
- Gravitational Force (Weight): The weight of the sphere (mg) acts vertically downward.
- Components of Weight: On the incline, the force component acting down the slope is mg * sin(θ), and the normal force is mg * cos(θ).
Rolling Motion
- Moment of Inertia: The moment of inertia (I) for a solid sphere is (2/5) * m * r^2.
- Acceleration Relation: The linear acceleration (a) of the sphere is related to its angular acceleration (α) by a = r * α, where r is the radius of the sphere.
Applying Newton's Second Law
1. Translational Motion: The net force acting down the incline is:
F_net = mg * sin(30°) - f,
where f is the friction force.
2. Rotational Motion: The torque τ due to friction is:
τ = f * r = I * α.
3. Combining Equations:
- Using f = (2/5) * m * a (from the rolling condition) and substituting it into F_net gives:
m * a = mg * sin(30°) - (2/5) * m * a.
4. Solve for a:
- Rearranging yields:
(7/5) * m * a = mg * (1/2),
hence, a = (5/7) * g * (1/2).
Final Linear Acceleration
- The linear acceleration of the solid sphere rolling down the inclined plane at an angle of 30° is:
a = (5/14) * g, approximately equal to 1.25 m/s² when g = 9.81 m/s².
This means the sphere accelerates down the incline at this rate, considering both translational and rotational motion.
When a solid sphere rolls down an inclined plane, the forces acting on it include gravitational force, normal force, and friction. The key to finding the linear acceleration is to analyze these forces and the moment of inertia of the sphere.
Forces on the Sphere
- Gravitational Force (Weight): The weight of the sphere (mg) acts vertically downward.
- Components of Weight: On the incline, the force component acting down the slope is mg * sin(θ), and the normal force is mg * cos(θ).
Rolling Motion
- Moment of Inertia: The moment of inertia (I) for a solid sphere is (2/5) * m * r^2.
- Acceleration Relation: The linear acceleration (a) of the sphere is related to its angular acceleration (α) by a = r * α, where r is the radius of the sphere.
Applying Newton's Second Law
1. Translational Motion: The net force acting down the incline is:
F_net = mg * sin(30°) - f,
where f is the friction force.
2. Rotational Motion: The torque τ due to friction is:
τ = f * r = I * α.
3. Combining Equations:
- Using f = (2/5) * m * a (from the rolling condition) and substituting it into F_net gives:
m * a = mg * sin(30°) - (2/5) * m * a.
4. Solve for a:
- Rearranging yields:
(7/5) * m * a = mg * (1/2),
hence, a = (5/7) * g * (1/2).
Final Linear Acceleration
- The linear acceleration of the solid sphere rolling down the inclined plane at an angle of 30° is:
a = (5/14) * g, approximately equal to 1.25 m/s² when g = 9.81 m/s².
This means the sphere accelerates down the incline at this rate, considering both translational and rotational motion.
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An inclined plane makes an angle of 30° with the horizontal. A solid sphere rolling down on this inclined plane from rest without slipping, has a linear acceleration, equal to?
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An inclined plane makes an angle of 30° with the horizontal. A solid sphere rolling down on this inclined plane from rest without slipping, has a linear acceleration, equal to? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about An inclined plane makes an angle of 30° with the horizontal. A solid sphere rolling down on this inclined plane from rest without slipping, has a linear acceleration, equal to? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for An inclined plane makes an angle of 30° with the horizontal. A solid sphere rolling down on this inclined plane from rest without slipping, has a linear acceleration, equal to?.
An inclined plane makes an angle of 30° with the horizontal. A solid sphere rolling down on this inclined plane from rest without slipping, has a linear acceleration, equal to? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about An inclined plane makes an angle of 30° with the horizontal. A solid sphere rolling down on this inclined plane from rest without slipping, has a linear acceleration, equal to? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for An inclined plane makes an angle of 30° with the horizontal. A solid sphere rolling down on this inclined plane from rest without slipping, has a linear acceleration, equal to?.
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