A 5 m long ladder is resting on a smooth vertical wall with its lower end 3 m from the wall. What should be the coefficient of friction between the ladder and the floor for equilibrium?

To solve this problem, we need to use the principles of static equilibrium and the concept of friction.

Step 1: Identify the forces acting on the ladder. There are three forces acting on the ladder: the weight of the ladder acting downwards from its center of mass, the normal force from the wall acting horizontally at the top of the ladder, and the normal force from the floor acting upwards at the bottom of the ladder.

Step 2: Set up the equilibrium conditions. For the ladder to be in equilibrium, the sum of the forces and the sum of the torques (moments) acting on it must be zero.

Step 3: Calculate the weight of the ladder. The weight of the ladder acts at its center of mass, which is at the midpoint of the ladder, i.e., 2.5 m from either end.

Step 4: Set up the equation for the sum of the torques. Taking torques about the bottom of the ladder, the torque due to the weight of the ladder is (Weight * 2.5 m), and the torque due to the normal force from the wall is (Normal force * 5 m). Setting these equal gives: Weight * 2.5 m = Normal force * 5 m.

Step 5: Solve for the normal force from the wall. From the above equation, we find that the normal force from the wall is (Weight * 2.5 m) / 5 m = Weight / 2.

Step 6: Set up the equation for the sum of the forces. The sum of the forces in the horizontal direction gives: Frictional force = Normal force from the wall. The frictional force is also equal to the coefficient of friction times the normal force from the floor, which is equal to the weight of the ladder.

Step 7: Solve for the coefficient of friction. Setting these equal gives: Coefficient of friction * Weight = Weight / 2. Solving for the coefficient of friction gives: Coefficient of friction = 1 / 2.

So, the coefficient of friction between the ladder and the floor should be 0.5 for the ladder to be in equilibrium.