A horizontal force F~ is applied to a block of mass m = 1 kg placed on an inclined at θ = 30◦ plane. The coefficients of static and dynamic friction are µs = 0.3 and µ = 0.2, respectively. What would be the minimum and maximum F such that the block is at rest?
To determine the minimum and maximum horizontal force (F) required to keep the block at rest on the inclined plane, we need to consider the forces acting on the block.
Let's break down the forces in the x and y directions:
In the x-direction:
- The force of gravity (mg) can be resolved into two components:
- mg*sin(θ) acts parallel to the inclined plane, in the downwards direction.
- mg*cos(θ) acts perpendicular to the inclined plane.
- The friction force (F_friction) acts opposite to the direction of motion (or intended motion) and has a maximum value of µs*N.
- N is the normal force, which is equal to mg*cos(θ) for this scenario.
- The maximum static friction force is given by µs*N = µs*mg*cos(θ), where µs is the coefficient of static friction.
In the y-direction:
- The normal force (N) acts perpendicular to the inclined plane and is equal to mg*cos(θ).
Now, let's analyze these forces to determine the minimum and maximum values of F:
Minimum Force (F_min):
- When the block is at rest, the applied force (F) is equal to the friction force (F_friction) in the opposite direction. So, F_min = F_friction.
- F_friction reaches its maximum value just before the block begins to move, which is when the force of static friction is overcome.
- Therefore, F_min = µs*mg*cos(θ).
Maximum Force (F_max):
- If we want to find the maximum force the block can withstand before it starts accelerating, we need to consider the dynamic (kinetic) friction force.
- In this case, F_max is the applied force required to overcome the kinetic friction force.
- The kinetic friction force is given by F_friction = µ*mg*cos(θ), where µ is the coefficient of kinetic friction.
- Therefore, F_max = µ*mg*cos(θ).
Now, plugging in the given values:
- m = 1 kg
- θ = 30° (radians = π/6)
- µs = 0.3 (coefficient of static friction)
- µ = 0.2 (coefficient of kinetic friction)
- g = 9.8 m/s² (acceleration due to gravity)
Calculating F_min:
F_min = µs*mg*cos(θ)
F_min = (0.3)(1 kg)(9.8 m/s²)(cos(π/6))
F_min ≈ 2.439 N
Calculating F_max:
F_max = µ*mg*cos(θ)
F_max = (0.2)(1 kg)(9.8 m/s²)(cos(π/6))
F_max ≈ 1.626 N
Hence, the minimum horizontal force required to keep the block at rest is approximately 2.439 N, while the maximum force the block can withstand before it starts accelerating is approximately 1.626 N.