The weightless strut in the figure below is not attached to the wall; it is prevented from falling only by friction. (Let w = 360 N, L = 5.25 m and h = 3.25 m.)
(a) Find the magnitude of the force of friction between the wall and the strut.
(b) Find the normal force exerted by the wall on the strut.
(c) Find the minimum coefficient of static friction.
To solve this problem, we can use the following equations:
(a) Magnitude of the force of friction (F_friction):
F_friction = μ * N
where μ is the coefficient of friction and N is the normal force.
(b) Normal force (N):
N = F_weight - F_parallel
where F_weight is the weight of the strut and F_parallel is the parallel force acting on the strut.
(c) Minimum coefficient of static friction (μ_static):
μ_static = tan(θ)
where θ is the angle between the weight of the strut and the horizontal axis.
Let's calculate the answers step-by-step:
(a) Magnitude of the force of friction (F_friction):
Since there is no vertical motion, the sum of the vertical forces should be equal to zero.
ΣF_y = N - F_weight = 0
N = F_weight
Substituting the given values:
N = 360 N
Now we can find the coefficient of friction:
F_friction = μ * N
Since the strut is prevented from falling, the force of friction should balance the horizontal force acting on the strut.
F_parallel = F_weight * sin(θ) = 360 N * sin(θ)
F_friction = F_parallel
μ * N = F_parallel
μ * 360 N = 360 N * sin(θ)
μ = sin(θ)
Substituting the given values:
μ = sin(3.25 m / 5.25 m)
μ ≈ 0.619
Therefore, the magnitude of the force of friction between the wall and the strut is approximately 0.619 times the weight of the strut.
(b) Normal force (N):
N = F_weight - F_parallel
Substituting the given values:
N = 360 N - 360 N * sin(θ)
N = 360 N * (1 - sin(θ))
N ≈ 360 N * (1 - sin(3.25 m / 5.25 m))
N ≈ 126.52 N
Therefore, the normal force exerted by the wall on the strut is approximately 126.52 N.
(c) Minimum coefficient of static friction (μ_static):
μ_static = tan(θ)
Substituting the given values:
μ_static = tan(3.25 m / 5.25 m)
μ_static ≈ 0.987
Therefore, the minimum coefficient of static friction is approximately 0.987.
To solve this problem, we can break it down into three parts: finding the force of friction, finding the normal force, and finding the minimum coefficient of static friction.
(a) Finding the magnitude of the force of friction:
The force of friction is given by the equation F_friction = μ * N, where μ is the coefficient of friction and N is the normal force. In this case, the force of friction is preventing the strut from falling, so it will be equal to the weight of the strut.
Since the strut is weightless, the weight of the strut is equal to the force acting downwards on it, which is 360 N. Therefore, the force of friction is also 360 N.
(b) Finding the normal force:
The normal force is the force exerted by a surface perpendicular to the surface. In this case, it is the force exerted by the wall on the strut.
The normal force can be found using the equation ∑Fy = 0, where ∑Fy represents the sum of the forces in the y direction. Since the strut is not accelerating in the y direction, the sum of the forces in the y direction must be zero.
There are two forces acting in the y direction: the weight of the strut acting downwards and the normal force acting upwards. Therefore, we have:
N - 360 N = 0
N = 360 N
(c) Finding the minimum coefficient of static friction:
The coefficient of static friction (μ_s) is defined as the ratio of the maximum force of static friction to the normal force.
In this case, the maximum force of static friction is equal to the force of friction (since the strut is not moving) and the normal force is 360 N. Therefore, we have:
μ_s = F_friction / N
μ_s = 360 N / 360 N
μ_s = 1
Therefore, the minimum coefficient of static friction is 1.