A uniform ladder of mass m is 2.5 m long. Its upper end leans against a frictionless wall. The coefficient of static friction between the ladder and the floor is 0.58. How far from the wall can the lower end be placed before the ladder starts to slip?

Hint: for the ladder to be in equilibrium, the sum of all forces must be zero, and the sum of all torques must be zero.

To determine the distance from the wall at which the ladder starts to slip, we need to consider the forces and torques acting on the ladder.

First, let's consider the forces. There are three main forces acting on the ladder:
1. The weight of the ladder acting downward, which is given by mg, where m is the mass of the ladder and g is the acceleration due to gravity.
2. The normal force exerted by the floor acting upward, which balances the weight of the ladder.
3. The frictional force acting parallel to the floor, between the ladder and the floor.

Since the ladder is on the verge of slipping, the maximum static frictional force (fs) will be acting in the opposite direction to prevent slipping. The maximum static frictional force can be determined using the coefficient of static friction (μs) and the normal force (N) as follows: fs = μs * N.

Now, let's consider the torques. The torque is the rotational force around an axis. In this case, the axis of rotation is the point where the ladder contacts the floor.

When the ladder is about to slip, the torque due to the weight of the ladder and the normal force must balance out the torque due to the frictional force acting at the bottom end of the ladder.

To calculate the torque, we need to choose a reference point. It is convenient to choose the point where the ladder contacts the floor as the reference point.

For the weight of the ladder, the torque is zero since the weight acts through the reference point.

For the normal force, the torque is zero since the normal force acts perpendicularly to the lever arm and creates no rotational force.

For the frictional force, the torque is given by the product of the frictional force (fs) and the lever arm (d), which is the distance between the reference point and the point where the frictional force acts.

Since the ladder is on the verge of slipping, the torque due to the frictional force must balance the torque due to the weight of the ladder.

Now, to solve for the distance from the wall at which the ladder starts to slip:
1. Express the equation for the sum of torques around the reference point.
2. Set up the equation for the sum of forces in the vertical direction (since the ladder is in equilibrium).
3. Use the equation for the maximum static frictional force to replace the frictional force term in the torque equation.
4. Solve the equations simultaneously to find the unknown distance.

By following these steps, you should be able to find the distance from the wall at which the ladder starts to slip.