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A uniform sphere of mass m and radius R rolls without slipping down an inclined plane set at an angle thetha to the horizontal. Find the friction coefficient at which slipping is absent?
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A uniform sphere of mass m and radius R rolls without slipping down an...
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A uniform sphere of mass m and radius R rolls without slipping down an...
Analysis:
The key to solving this problem is to understand the conditions for rolling without slipping on an inclined plane.
Forces Acting:
1. The gravitational force mg acting downwards.
2. The normal force N perpendicular to the plane.
3. The frictional force f opposing the motion.
Torque Equilibrium:
For the sphere to roll without slipping, the torque about the center of mass must be zero.
Frictional Force:
The frictional force can be expressed as f = μN, where μ is the coefficient of friction.
Applying Equations:
1. ΣFy = N - mgcos(θ) = 0 (vertical equilibrium)
2. ΣFx = f - mgsin(θ) = 0 (horizontal equilibrium)
3. τ = Iα = 0 (torque equilibrium)
Substitute Values:
1. N = mgcos(θ)
2. f = mgsin(θ)
3. α = a/R
Find the Coefficient of Friction:
Substitute the expressions for N and f into the equation for the frictional force.
Final Result:
The coefficient of friction at which slipping is absent is μ = tan(θ).
The key to solving this problem is to understand the conditions for rolling without slipping on an inclined plane.
Forces Acting:
1. The gravitational force mg acting downwards.
2. The normal force N perpendicular to the plane.
3. The frictional force f opposing the motion.
Torque Equilibrium:
For the sphere to roll without slipping, the torque about the center of mass must be zero.
Frictional Force:
The frictional force can be expressed as f = μN, where μ is the coefficient of friction.
Applying Equations:
1. ΣFy = N - mgcos(θ) = 0 (vertical equilibrium)
2. ΣFx = f - mgsin(θ) = 0 (horizontal equilibrium)
3. τ = Iα = 0 (torque equilibrium)
Substitute Values:
1. N = mgcos(θ)
2. f = mgsin(θ)
3. α = a/R
Find the Coefficient of Friction:
Substitute the expressions for N and f into the equation for the frictional force.
Final Result:
The coefficient of friction at which slipping is absent is μ = tan(θ).
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A uniform sphere of mass m and radius R rolls without slipping down an inclined plane set at an angle thetha to the horizontal. Find the friction coefficient at which slipping is absent?
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A uniform sphere of mass m and radius R rolls without slipping down an inclined plane set at an angle thetha to the horizontal. Find the friction coefficient at which slipping is absent? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about A uniform sphere of mass m and radius R rolls without slipping down an inclined plane set at an angle thetha to the horizontal. Find the friction coefficient at which slipping is absent? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A uniform sphere of mass m and radius R rolls without slipping down an inclined plane set at an angle thetha to the horizontal. Find the friction coefficient at which slipping is absent?.
A uniform sphere of mass m and radius R rolls without slipping down an inclined plane set at an angle thetha to the horizontal. Find the friction coefficient at which slipping is absent? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about A uniform sphere of mass m and radius R rolls without slipping down an inclined plane set at an angle thetha to the horizontal. Find the friction coefficient at which slipping is absent? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A uniform sphere of mass m and radius R rolls without slipping down an inclined plane set at an angle thetha to the horizontal. Find the friction coefficient at which slipping is absent?.
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