A ladder rests against a smooth wall (by smooth wall, we mean that the wall exerts in the ladder only a force that is perpendicular to the wall. There is no friction force). The ladder weighs 200N and its center of gravity is 0.4L from the base, where L is the ladder’s length. Find
How large a friction force must exist at the base of the ladder if it is not to slip?
What is the necessary coefficient of static friction
Ladder is slip and men is die
To find the required friction force at the base of the ladder and the coefficient of static friction, we can use the concept of torque and equilibrium.
Step 1: Determine the forces acting on the ladder.
- Weight of the ladder (200N) acts vertically downward from its center of gravity, which is 0.4L from the base.
- Normal force (N) acts perpendicular to the smooth wall and balances the weight of the ladder.
- Friction force (F) acts at the base of the ladder and prevents it from slipping.
Step 2: Set up the torque equation.
Since the ladder is in equilibrium (no rotational motion), the sum of the torques about any point must be zero. We can choose the base of the ladder as the reference point for convenience.
The torque due to the weight of the ladder about the base is given by:
Torque = Weight * Perpendicular distance = 200N * (0.4L)
The torque due to the friction force about the base is given by:
Torque = Friction force * Distance of the base from the reference point = F * 0
Since the ladder does not slip, the friction force creates a torque in the opposite direction to the torque from the weight of the ladder. Therefore, the torques must be equal in magnitude:
200N * (0.4L) = F * 0
Step 3: Solve for the friction force.
Since the distance of the base from the reference point is zero and any number multiplied by zero is zero, the torque equation simplifies to:
0 = 0
This means that the friction force can have any value, as long as it prevents the ladder from slipping. So, there is no specific value for the friction force.
Step 4: Determine the coefficient of static friction.
The coefficient of static friction (μs) is the ratio of the maximum static friction force to the normal force. In this case, there is no specific value for the friction force, so the coefficient of static friction can also have any value between 0 and infinity.
To summarize, the friction force at the base of the ladder can have any value as long as it prevents slipping. Similarly, the coefficient of static friction can have any value between 0 and infinity.