A uniform ladder, 3 m long and weighing 130 N, placed against a smooth wall, its lower end being 1.80 m from the wall. A 700N man stands on the upper most rung of the ladder located 0.45 m from the upper end of the ladder. a.) What are the vertical and horizontal forces exerted by the ground on the ladder? b.) What is the coefficient of friction between the ladder and the ground?
draw the diagram. AT the uppermost contant point, only a normal force into the will exists. At the bottom, both normal, and horizontal (friction) exists.
at the center of the ladder, its weight force exists, downward. The man has weight at his center of gravity, act downward.
Sum the vertical forces, they equal zero.
Sum the horizntal forces,they sum to zero.
sum the moments about any point, I recommned the top of the ladder.
you have three unknowns (fh at top, fh at bottom (friction), and fv at the bottom) and you have three equations, so it is solvable.
174N&40N
To find the vertical and horizontal forces exerted by the ground on the ladder, we need to consider the forces acting on the ladder in equilibrium.
Let's assume the vertical force exerted by the ground on the ladder is Fv and the horizontal force exerted by the ground on the ladder is Fh.
a.) To find the vertical force (Fv), we need to balance the forces in the vertical direction:
The vertical forces acting on the ladder are:
1. The weight of the ladder (130 N) acting downwards.
2. The weight of the man (700 N) acting downwards.
Since the ladder is in equilibrium, the net vertical force should be zero. So, we have:
Fv - 130 N - 700 N = 0
Simplifying the equation, we find:
Fv = 830 N
So, the vertical force exerted by the ground on the ladder is 830 Newtons.
b.) To find the coefficient of friction between the ladder and the ground, we need to consider the horizontal forces:
The horizontal forces acting on the ladder are:
1. The horizontal component of the weight of the ladder (130 N) acting towards the wall.
2. The horizontal component of the weight of the man (700 N) acting towards the wall.
3. The horizontal force exerted by the wall on the ladder.
Since the ladder is in equilibrium, the net horizontal force should be zero. So, we have:
Fh - 130 N - 700 N - F_wall = 0
Simplifying the equation, we find:
Fh = 830 N + F_wall
Now, we need to find the horizontal force exerted by the wall on the ladder. This force can be calculated using the torque equation:
Torque = Force x Distance
The torque caused by the weight of the ladder can be calculated as:
Torque_ladder = Weight_ladder x Distance_ladder = 130 N x (3 m - 1.80 m)
The torque caused by the weight of the man can be calculated as:
Torque_man = Weight_man x Distance_man = 700 N x (0.45 m)
Since the ladder is in rotational equilibrium, the torque caused by these forces must be equal and opposite to the torque exerted by the wall:
Torque_ladder + Torque_man = F_wall x Distance_wall
(130 N x (3 m - 1.80 m)) + (700 N x (0.45 m)) = F_wall x 1.80 m
Solving for F_wall, we find:
F_wall = [(130 N x (3 m - 1.80 m)) + (700 N x (0.45 m))] / 1.80 m
Calculating F_wall, we get:
F_wall ≈ 843.33 N
Now, substituting the value of F_wall in the equation for Fh, we have:
Fh = 830 N + 843.33 N
Fh ≈ 1673.33 N
So, the horizontal force exerted by the ground on the ladder is approximately 1673.33 Newtons.
To calculate the coefficient of friction between the ladder and the ground, we can use the equation:
Coefficient of friction = Fh / Fv
Coefficient of friction = 1673.33 N / 830 N
Coefficient of friction ≈ 2.02
Therefore, the coefficient of friction between the ladder and the ground is approximately 2.02.