A ladder has a weight of 400 N and a length of 10.0 m, it is plac s over a wall without friction. A person whose wight is of 800N stands in the ladder and is 2m from the bottom of the ladder, which is 8 m over the inferior part of the wall.
Calculate the force that the wall exerts over the ladder and the normal force exerted by the floor.
To find the force that the wall exerts on the ladder and the normal force exerted by the floor, we can analyze the forces acting on the ladder.
First, let's calculate the torque exerted by the person standing on the ladder. Torque is given by the formula:
Torque = force × lever arm
The force exerted by the person is their weight, which is 800 N, and the lever arm is the distance from the person to the bottom of the ladder, which is 2 m. Therefore, the torque exerted by the person is:
Torque = 800 N × 2 m = 1600 N·m
Next, let's calculate the torque due to the weight of the ladder. The weight of the ladder is 400 N, and the distance from the center of mass of the ladder to the bottom of the ladder is half its length, so it is 5 m. Therefore, the torque due to the weight of the ladder is:
Torque = 400 N × 5 m = 2000 N·m
Since the ladder is in equilibrium (not rotating), the torque exerted by the person must be equal to the torque due to the weight of the ladder. Therefore:
1600 N·m = 2000 N·m
Now, to calculate the force that the wall exerts on the ladder, we can use the fact that the torque exerted by the wall is equal in magnitude and opposite in direction to the torque exerted by the person. Hence, the torque exerted by the wall is 1600 N·m.
The distance from the point where the ladder contacts the wall to the bottom of the ladder is 8 m. Therefore, we can calculate the force that the wall exerts on the ladder using the equation for torque:
Torque = force × lever arm
1600 N·m = force × 8 m
Solving for the force, we find:
force = 1600 N·m / 8 m = 200 N
So, the force that the wall exerts on the ladder is 200 N.
Finally, to find the normal force exerted by the floor, we can use the fact that the sum of the vertical forces must be equal to zero, as the ladder is in equilibrium. The vertical forces acting on the ladder are the weight of the person and the weight of the ladder, balanced by the normal force exerted by the floor:
Weight of person + Weight of ladder = Normal force
800 N + 400 N = Normal force
Normal force = 1200 N
Therefore, the normal force exerted by the floor is 1200 N.
To solve this problem, we can analyze the forces acting on the ladder and the person standing on it.
1. First, let's consider the ladder's force equilibrium. The ladder is in rotational equilibrium, meaning the net torque acting on it is zero.
The torque created by the person's weight (800 N) about the bottom of the ladder (2 m from the bottom) is given by the formula τ = F * d * sin(θ), where F is the force, d is the distance from the pivot point, and θ is the angle between the force and the lever arm. In this case, the angle θ is 90 degrees, so sin(θ) = 1.
τ = (800 N) * (2 m) * (1) = 1600 N·m
Since the torque is zero, we know that the torque due to the ladder's weight must balance the torque due to the person's weight.
τ_ladder = (Weight of ladder) * (distance to pivot point)
τ_ladder = (400 N) * (5 m) = 2000 N·m
As the torques are equal:
2000 N·m = 1600 N·m + F_wall * (8 m)
Solving for F_wall (force exerted by the wall):
F_wall = (2000 N·m - 1600 N·m) / 8 m
F_wall = 400 N
Therefore, the force exerted by the wall on the ladder is 400 N.
2. Next, let's determine the normal force exerted by the floor on the ladder.
Considering the vertical forces, we can find the normal force exerted by the floor.
Summing up the forces in the vertical direction:
ΣF_y = 0
N_floor - (Weight of ladder) - (Person's weight) = 0
N_floor = (Weight of ladder) + (Person's weight)
N_floor = 400 N + 800 N
N_floor = 1200 N
Therefore, the normal force exerted by the floor on the ladder is 1200 N.