A block of a mass of 1 kg rests on the inclined surface of a wedge of a mass of 2.2 kg. The wedge is acted upon by a horizontal force F and slides on a frictionless surface. The acceleration of gravity is 9.81 m/s^2.
If the coefficient of static friction between the wedge and the block is .6 and the angle of the incline is 36 degrees,find the minimum value of F for which the block does not slip. Answer in N.
FInd the maximum value of F for which the block does not slip. Answer in N.
To find the minimum and maximum values of F for which the block does not slip, we need to analyze the forces acting on the block and the wedge.
Let's start by considering the forces acting on the block. There are two forces acting on the block: the gravitational force (mg) and the normal force (N) exerted by the wedge. The normal force can be resolved into two components: perpendicular to the incline (N⊥) and parallel to the incline (N∥). The block will start to slip if the parallel force (N∥) exceeds the static friction force (fs).
Now, let's calculate the normal force (N) and the static friction force (fs).
1. Normal force (N):
The normal force (N) can be found by considering the vertical equilibrium of the block. Since the block is not moving vertically, the sum of the forces in the vertical direction must be zero.
N - mgcos(θ) = 0
N = mgcos(θ)
N = 1 kg * 9.81 m/s^2 * cos(36°)
N ≈ 7.882 N
2. Static friction force (fs):
The static friction force (fs) can be calculated using the equation:
fs ≤ μsN
where μs is the coefficient of static friction between the wedge and the block, and N is the normal force.
fs ≤ 0.6 * 7.882 N
fs ≤ 4.7292 N
Now, let's analyze the forces acting on the wedge. Since the surface is frictionless, the only force acting on the wedge is the force F applied horizontally.
3. Reaction force on the wedge:
The reaction force on the wedge (R) can be calculated using the equation:
R = mg + N⊥
R = 2.2 kg * 9.81 m/s^2 + N * sin(θ)
R = 2.2 kg * 9.81 m/s^2 + 7.882 N * sin(36°)
R ≈ 32.458 N + 4.7412 N
R ≈ 37.199 N
Now, let's calculate the minimum and maximum values of F.
Minimum value of F (to prevent slipping):
To prevent slipping, the parallel force (N∥) exerted by the wedge must equal the static friction force (fs).
N∥ = fs
F = N∥
F = fs = 4.7292 N
Maximum value of F (to prevent slipping):
To find the maximum value of F, we need to consider the forces acting on the block and the wedge.
The maximum value of F would be when the parallel force (N∥) exerted by the wedge is at its maximum, which occurs when the block is about to slip.
N∥ = fs = 4.7292 N
Now, let's calculate the maximum value of F.
N∥ = F - R
4.7292 N = F - 37.199 N
F = 41.9282 N
Therefore, the minimum value of F for which the block does not slip is approximately 4.7292 N, and the maximum value of F for which the block does not slip is approximately 41.9282 N.