A person using a ladder supported on vertical wall is 3/4 of the way up the ladder. If the person climbing the ladder has a weight of 980 newtons and the ladder is 4.89 meters long, how far from the wall can the base of the ladder be placed, and not slip? The coefficient of friction between the base of the ladder and the ground is 0.34. Assume that there is no friction between the ladder and the wall and that the ladder is effectively weightless.
To solve this problem, we need to consider the forces acting on the ladder. Let's break it down step by step:
1. Find the weight of the person on the ladder:
The weight of the person is given as 980 newtons.
2. Find the weight of the ladder:
Since the ladder is effectively weightless, its weight is negligible.
3. Find the force exerted by the person along the ladder:
The force exerted along the ladder is equal to the weight of the person. So, the force exerted by the person along the ladder is 980 newtons.
4. Determine the force of friction between the ladder and the ground:
The force of friction between two surfaces can be calculated using the equation: Force of friction = coefficient of friction × normal force.
Since the ladder is at rest, the sum of the vertical forces acting on it must be zero. The normal force is equal to the weight of the person (980 N). Therefore, the force of friction between the ladder and the ground is given by: Force of friction = 0.34 × 980 = 333.2 newtons.
5. Find the distance of the center of mass of the ladder from the base of the ladder:
Since the person is 3/4 of the way up the ladder, the center of mass of the ladder will also be 3/4 of the ladder's length from the base.
Distance of center of mass (x) = (3/4) × 4.89 = 3.67 meters.
6. Find the torque exerted by the person around the base of the ladder:
The torque exerted by the person around the base of the ladder can be calculated using the equation: Torque = force × distance.
Torque exerted by the person (τ) = 980 N × 3.67 m = 3596.6 N·m.
7. Determine the torque due to the force of friction:
The torque due to the force of friction can be calculated using the equation: Torque = force × distance.
Torque due to friction (τ_friction) = 333.2 N × x (unknown distance from the base).
8. Equate the torques to find the unknown distance from the base:
Since the ladder is in equilibrium, the sum of the torques around any point must be zero.
Therefore, we can equate the torque exerted by the person and the torque due to friction: τ = τ_friction.
3596.6 N·m = 333.2 N × x.
Solving for x, we find: x = 3596.6 N·m / 333.2 N ≈ 10.8 meters.
Therefore, the base of the ladder can be placed approximately 10.8 meters from the wall to prevent slipping.