UPSC Exam > UPSC Questions > An inclined plane makes an angle of 30° with ...
Start Learning for Free
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?
Most Upvoted Answer
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.
Attention UPSC Students!
To make sure you are not studying endlessly, EduRev has designed UPSC study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in UPSC.
|
Explore Courses for UPSC exam
|
|
Similar UPSC Doubts
Top Courses for UPSCView all
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?
Question Description
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?.
Solutions 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? in English & in Hindi are available as part of our courses for UPSC.
Download more important topics, notes, lectures and mock test series for UPSC Exam by signing up for free.
Here you can find the meaning of 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? defined & explained in the simplest way possible. Besides giving the explanation of
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?, a detailed solution 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? has been provided alongside types of 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? theory, EduRev gives you an
ample number of questions to practice 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? tests, examples and also practice UPSC tests.
|
|
Explore Courses for UPSC exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup