A block is sliding down a frictionless incline of angle θ. If the incline has a coefficient of static friction μs, what is the maximum angle θ for which the block will not slide?
Practice Questions
1 question
Q1
A block is sliding down a frictionless incline of angle θ. If the incline has a coefficient of static friction μs, what is the maximum angle θ for which the block will not slide?
tan⁻¹(μs)
sin⁻¹(μs)
cos⁻¹(μs)
μs
The block will not slide if the component of gravitational force down the incline is less than or equal to the maximum static friction force, leading to θ = tan⁻¹(μs).
Questions & Step-by-step Solutions
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Q
Q: A block is sliding down a frictionless incline of angle θ. If the incline has a coefficient of static friction μs, what is the maximum angle θ for which the block will not slide?
Solution: The block will not slide if the component of gravitational force down the incline is less than or equal to the maximum static friction force, leading to θ = tan⁻¹(μs).
Steps: 11
Step 1: Understand that the block is on an incline and can slide down due to gravity.
Step 2: Identify the forces acting on the block: gravitational force and static friction force.
Step 3: The gravitational force can be split into two components: one acting down the incline and one acting perpendicular to the incline.
Step 4: The component of gravitational force down the incline is given by mg sin(θ), where m is the mass of the block and g is the acceleration due to gravity.
Step 5: The maximum static friction force that can act up the incline is given by μs * N, where N is the normal force.
Step 6: The normal force N is equal to mg cos(θ) because it acts perpendicular to the incline.
Step 7: Therefore, the maximum static friction force is μs * mg cos(θ).
Step 8: For the block to not slide, the gravitational force down the incline must be less than or equal to the maximum static friction force: mg sin(θ) ≤ μs * mg cos(θ).
Step 9: Cancel mg from both sides of the inequality (assuming m is not zero): sin(θ) ≤ μs cos(θ).
Step 10: Rearranging gives us tan(θ) ≤ μs.
Step 11: To find the maximum angle θ, we take the arctangent: θ = tan⁻¹(μs).
