An 8.5 m ladder whose weight is 310 N is placed against a smooth vertical wall. A person whose weight is 515 N stands on the ladder a distance 1.8 m up the ladder. The foot of the ladder rests on the floor 5.78 m from the wall.
Calculate the normal force exerted by the floor on the ladder.
Answer in units of N
The only vertical force on the person-ladder system is the force up from the floor.
Therefore this vertical force up must balance the weight of the person plus the ladder.
Fup = 310 + 515 = 825 N
Now if you asked for the HORIZONTAL force from the wall on the ladder, then we would have had to do moments.
To calculate the normal force exerted by the floor on the ladder, we need to consider the forces acting on the ladder.
First, let's consider the forces acting vertically. The weight of the ladder itself, 310 N, acts downward at its center of mass. The weight of the person, 515 N, acts downward at the position of the person on the ladder. The normal force exerted by the floor acts upward and is the force we need to calculate.
Next, let's consider the forces acting horizontally. Since the ladder is on a smooth vertical wall, it means there is no frictional force between the ladder and the wall. Therefore, the only horizontal force is the reaction force between the floor and the ladder at the point where the ladder rests on the floor.
Now, let's find the horizontal reaction force. We can use the torque equation to find the horizontal reaction force.
Torque = Force x Distance
The torque created by the weight of the ladder is zero because it acts along the line of action of the force. Therefore, the only torque acting on the ladder is the torque created by the weight of the person.
Torque due to person's weight = Person's weight x Distance from the person to the wall
Torque due to person's weight = 515 N x 1.8 m
Torque due to person's weight = 927 N·m
Since the ladder is in equilibrium, the torque due to the horizontal reaction force exerted by the floor must be equal and opposite to the torque due to the person's weight.
Torque due to horizontal reaction force = Torque due to person's weight
Horizontal reaction force x Distance from the ladder's center of mass to the wall = 927 N·m
Horizontal reaction force = 927 N·m / 5.78 m
Horizontal reaction force = 160.55 N
However, since the ladder is smooth and there is no horizontal force acting on it, the horizontal reaction force must be zero. Therefore, the normal force exerted by the floor on the ladder is equal to the horizontal reaction force, which is zero.
So, the normal force exerted by the floor on the ladder is 0 N.