To determine the wall's reaction force on the ladder, we need to analyze the forces acting on the ladder and apply the principles of static equilibrium. In this scenario, the ladder is leaning against a frictionless wall, which means that the only horizontal force acting on it is the wall's reaction force. Let's break this down step by step.
Understanding the Setup
We have a ladder leaning against a wall, with the following details:
- Height of the upper end above the ground: 6 m
- Distance of the lower end from the wall: 4 m
- Weight of the ladder: 500 N
- Center of gravity located at 1/3 of the ladder's length from the lower end
Finding the Length of the Ladder
First, we can find the length of the ladder using the Pythagorean theorem. The ladder forms a right triangle with the wall and the ground:
Let \( L \) be the length of the ladder. According to the Pythagorean theorem:
L² = height² + base²
Substituting the values:
L² = 6² + 4² = 36 + 16 = 52
Thus, the length of the ladder is:
L = √52 ≈ 7.21 m
Locating the Center of Gravity
The center of gravity is located at 1/3 of the ladder's length from the lower end:
Distance from the lower end to the center of gravity = (1/3) * 7.21 m ≈ 2.40 m
Analyzing Forces and Moments
In static equilibrium, the sum of vertical forces and the sum of moments about any point must be zero. The forces acting on the ladder include:
- The weight of the ladder (500 N) acting downward at the center of gravity.
- The wall's reaction force (R) acting horizontally at the top of the ladder.
- The ground's reaction force (N) acting vertically upward at the base of the ladder.
Setting Up the Equations
1. **Vertical Forces**: The sum of vertical forces must equal zero:
N - 500 N = 0
Thus, N = 500 N.
2. **Moments About the Base**: To find the wall's reaction force, we can take moments about the base of the ladder. The moment due to the weight of the ladder is:
Moment = Weight × Distance from the base = 500 N × 2.40 m = 1200 Nm (clockwise)
The moment due to the wall's reaction force is:
Moment = R × Height of the ladder = R × 6 m (counterclockwise)
Setting the Moments Equal
For equilibrium, the clockwise moments must equal the counterclockwise moments:
1200 Nm = R × 6 m
Solving for R gives:
R = 1200 Nm / 6 m = 200 N
Final Result
The wall's reaction force acting on the ladder is 200 N. This force is crucial for maintaining the ladder's equilibrium against the frictionless wall, allowing it to remain stable while supporting its weight.
