prove that coefficient of static friction is tangent of angle of repose
Answers
Answer :
The coefficient of static friction (µs) is the ratio of the maximum frictional force to the normal force pressing two surfaces together. The angle of repose is the maximum angle at which a pile of loose material can form a stable slope without collapsing. The relationship between µs and the angle of repose is established through the equation of static equilibrium, where the friction force is equal to the weight of the material times the tangent of the angle of repose (µs = tan(θ)).
Explanation :
The coefficient of static friction (µs) between two surfaces is defined as the ratio of the maximum frictional force that can be exerted between the surfaces to the normal force pressing the two surfaces together. The angle of repose, on the other hand, is defined as the maximum angle at which a pile of loose granular material, such as sand, can form a stable slope without collapsing.
The relationship between the coefficient of static friction and the angle of repose can be established through the equation of static equilibrium. If a pile of material is at rest on a horizontal surface, the force of friction acting on the material is equal and opposite to the force of gravity, which is pulling the material down the slope. The force of friction can be represented as:
friction = μs * normal force
where μs is the coefficient of static friction and normal force is equal to the weight of the material pressing down on the surface, given by:
normal force = weight * cos(θ)
where weight is the total weight of the material and θ is the angle between the horizontal surface and the slope of the material. Substituting this expression into the equation for friction, we get:
friction = μs * weight * cos(θ)
For the angle of repose, the friction force is equal to the force of gravity, meaning the pile is on the verge of collapsing. Setting friction equal to weight * sin(θ), we get:
μs * weight * cos(θ) = weight * sin(θ)
Dividing both sides by weight * cos(θ), we get:
μs = tan(θ)
This equation shows that the coefficient of static friction is equal to the tangent of the angle of repose. In other words, the angle of repose can be determined by measuring the coefficient of static friction and finding the inverse tangent of that value.
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