A 3.0-m-long ladder, as shown in Figure 12.35, leans against a frictionless wall. The coefficient of static friction between the ladder and the floor is 0.40. What is the minimum angle the ladder can make with the floor without slipping?

검증된 단계별 안내

Step 1: Begin by analyzing the forces acting on the ladder. The ladder is in static equilibrium, meaning the sum of all forces and torques acting on it must be zero. Identify the forces: (1) the gravitational force acting downward at the ladder's center of mass, (2) the normal force exerted by the floor, (3) the frictional force at the base of the ladder, and (4) the normal force exerted by the wall.

Step 2: Write the equations for static equilibrium. For forces, the sum of horizontal forces must be zero: \( F_{friction} = F_{wall} \). The sum of vertical forces must also be zero: \( F_{normal, floor} = F_{gravity} \). For torques, choose the base of the ladder as the pivot point and set the sum of torques to zero.

Step 3: Express the torque equation. The torque due to gravity is \( \tau_{gravity} = F_{gravity} \cdot \frac{L}{2} \cdot \cos(\theta) \), where \( L \) is the length of the ladder and \( \theta \) is the angle the ladder makes with the floor. The torque due to the wall's normal force is \( \tau_{wall} = F_{wall} \cdot L \cdot \sin(\theta) \). Set \( \tau_{gravity} = \tau_{wall} \).

Step 4: Relate the frictional force to the normal force using the coefficient of static friction: \( F_{friction} \leq \mu \cdot F_{normal, floor} \), where \( \mu \) is the coefficient of static friction. Substitute \( F_{friction} = F_{wall} \) and \( F_{normal, floor} = F_{gravity} \) into this inequality.

Step 5: Combine the torque equation and the friction inequality to solve for the minimum angle \( \theta \). Rearrange the equations to isolate \( \theta \), ensuring that the ladder does not slip. Use trigonometric relationships to express \( \tan(\theta) \) in terms of \( \mu \) and other known quantities.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

19m

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Static Friction

Static friction is the force that prevents an object from moving when it is at rest. It acts parallel to the surfaces in contact and is dependent on the normal force and the coefficient of static friction. In this scenario, the ladder's stability relies on the static friction between it and the floor, which must be sufficient to counteract the forces acting on the ladder.

Equilibrium Conditions

For the ladder to remain in equilibrium, the sum of forces and the sum of torques acting on it must be zero. This means that the upward force from the ground must balance the downward gravitational force, and the torque due to the weight of the ladder must be countered by the torque due to the frictional force at the base. Understanding these conditions is crucial for determining the minimum angle before slipping occurs.

Angle of Inclination

The angle of inclination refers to the angle formed between the ladder and the horizontal ground. This angle affects the distribution of forces acting on the ladder, including the normal force and the gravitational force. By analyzing how this angle influences the static friction and the forces at play, one can calculate the minimum angle required to prevent slipping.