Equilibrium of a Ladder Leaning Against a Wall: Key Forces Involved

When a ladder is leaning against a wall and is at rest, several forces must balance to achieve equilibrium. Let's explore the forces involved and which force must be equal to the frictional force exerted on the ladder by the floor.

Forces Involved

Consider a ladder leaning against a wall. The forces acting on the ladder are:

Weight of the Ladder: Acts vertically downward at the center of mass of the ladder. Normal Force from the Floor: Acts vertically upward, counteracting the weight of the ladder. Frictional Force: Acts horizontally at the base of the ladder, preventing it from sliding away from the wall. Normal Force from the Wall: Acts horizontally on the ladder, pushing it away from the wall.

Equilibrium Conditions

For the ladder to be in equilibrium, the forces must balance. This means that the horizontal and vertical components of the forces must counteract each other.

Horizontal Equilibrium

The horizontal component of the normal force exerted by the wall on the ladder (Rw) must balance the horizontal frictional force exerted by the floor (Fg) to prevent the ladder from sliding.

The horizontal force equation is:

Rw Fg μgRg

Vertical Equilibrium

The vertical forces must also balance. The normal force from the floor (Rg) and the component of the ladder's weight (w) that is vertical must be equal.

The vertical force equation is:

Rg w

Frictional Forces and Normal Forces

The key issue is understanding the relationship between the frictional force and the normal forces. The frictional force exerted by the floor on the ladder (Fg) is equal in magnitude to the horizontal component of the normal force exerted by the wall on the ladder (Rw).

In mathematical terms, if we denote the coefficient of friction between the ladder and the floor as μg, we have:

Fg μgRg

Since the normal force exerted by the wall (Rw) is equal to the horizontal component of the force exerted by the wall (which is the same as the frictional force required to keep the ladder stationary), we can write:

Rw Fg

Understanding the Forces

To better visualize these forces, one can draw a free body diagram. This diagram would show the ladder with the following forces:

The weight of the ladder acting vertically downward. The frictional force (Fg) from the wall acting upward. The normal force (Rg) from the floor acting upward. The frictional force (Fg) at the floor acting horizontally. The normal force (Rw) from the wall acting horizontally.

These forces must balance to ensure the ladder remains in equilibrium. The horizontal component of the normal force from the wall (Rw) must be equal to the frictional force at the floor (Fg) to prevent the ladder from sliding.

Conclusion

In conclusion, for a ladder leaning against a wall at rest, the force that must be equal to the frictional force exerted on the ladder by the floor is the normal force exerted by the wall on the ladder. This ensures the ladder remains in a state of static equilibrium.