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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).
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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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