uniform ladder of length 5m and weight 80 newtons stands on rough level ground and rests in equilibrium against a smooth horizontal rail which is fixed 4m vertically above the ground. If the inclination of the ladder to the vertical is x, where tan x is less than 3/4, find expressions in terms of x for the vertical reaction R of the ground, the friction F at the ground and the normal reaction N at the rail.
To solve this problem, we need to consider the forces acting on the ladder in equilibrium.
Let's analyze the forces:
1. The weight, W, acts downwards from the center of the ladder. W = 80 N.
2. The vertical reaction, R, acts upwards from the ground.
3. The normal reaction, N, acts downwards from the rail.
4. The friction force, F, acts horizontally from the ground.
Now, let's apply the conditions given in the problem:
1. The inclination of the ladder to the vertical is x, where tan x < 3/4.
This means that the angle x must be less than the arctan(3/4) ≈ 36.87 degrees.
Now, let's calculate the expressions in terms of x for the forces:
1. Vertical reaction, R:
Since the ladder is in equilibrium, the sum of the vertical forces on the ladder must be zero.
R + N - W = 0
R + N - 80 = 0
R + N = 80
2. Friction force, F:
The ladder is in equilibrium horizontally, which means the sum of the horizontal forces on the ladder must be zero.
F = 0 (no horizontal force acting on the ladder)
3. Normal reaction, N:
To find the expression of N in terms of x, we need to analyze the forces acting on the ladder along the vertical direction.
The ladder can be considered as a right-angled triangle, with the hypotenuse being the ladder itself.
Using trigonometry, we can write:
sin x = N / 5 (vertical component of the hypotenuse is N)
N = 5 * sin x
Therefore, the expressions in terms of x for the forces are:
R + N = 80
N = 5 * sin x
F = 0
To find the expressions for the vertical reaction R of the ground, the friction F at the ground, and the normal reaction N at the rail, we need to analyze the forces acting on the ladder.
Let's start by drawing a free-body diagram of the ladder:
1. The weight of the ladder acts vertically downward and can be measured as 80 N. We can represent it as W.
2. The normal reaction N at the rail acts perpendicular to the rail and opposes the weight of the ladder. It keeps the ladder in equilibrium, preventing it from sliding down the rail.
3. The vertical reaction R at the ground prevents the ladder from sinking into the ground. It acts perpendicular to the ground and opposes the weight of the ladder.
4. The friction force F acts parallel to the ground and opposes any tendency of the ladder to slide horizontally.
To find the expressions for R, F, and N, we can consider the equilibrium of moments about the point where the ladder touches the ground (considered as the pivot point).
Taking moments about this point:
Sum of anticlockwise moments = Sum of clockwise moments
W * 4m = R * 5m + N * 4m
Now, we have two unknowns in this equation (R and N). To find their separate expressions, we need one more equation relating them. This equation can be derived by considering the inclination of the ladder.
Given that tan(x) < 3/4, we can express tan(x) as the ratio of the vertical height (h) to the horizontal length (d):
tan(x) = h / d
The vertical height can be determined as the difference between the vertical position of the rail and the ground level:
h = 4m - d
Substituting this value of h in the equation, we get:
tan(x) = (4m - d) / d
Cross-multiplying and rearranging, we get:
d * tan(x) = 4m - d
Simplifying further, we have:
d(tan(x) + 1) = 4m
Now, we can solve this equation to express d in terms of x:
d = 4m / (tan(x) + 1)
Substituting this value of d in the moment equation, we get:
80N * 4m = R * 5m + N * (4m / (tan(x) + 1))
Now, we have expressions for R and N in terms of x:
R = (80N * 4m - N * (4m / (tan(x) + 1))) / 5m
N = (80N * 4m - R * 5m) / (4m / (tan(x) + 1))
Please note that in this system of equations, N and R are intertwined. To get specific values for N and R, you would need to know either one of them or substitute values for x and solve the equations numerically.