Question

A ladder (ℓL = 7.80 m) of weight WL = 350 N leans against a smooth vertical wall. The term "smooth" means that the wall can exert only a normal force directed perpendicular to the wall and cannot exert a frictional force parallel to it. A firefighter, whose weight is 885 N, stands 6.40 m up from the bottom of the ladder (this distance goes along the ladder, it is not the vertical height). Assume that the ladder's weight acts at the ladder's center, and neglect the hose's weight. What is the minimum value for the coefficient of static friction between the ladder and the ground, so that the ladder (with the fireman on it) does not slip? (Assume = 54.0°.)

Answer 1

Answer: In order for the ladder (with the firefighter on it) to not slip against the ground, the friction between the ladder and the ground must provide enough horizontal force to prevent the ladder from sliding horizontally. The minimum value of the coefficient of static friction between the ladder and the ground can be calculated using the following steps:

Step 1: Identify the forces acting on the ladder.

The forces acting on the ladder are the weight of the ladder (WL) acting downward at the center of the ladder, the weight of the firefighter (Wf) acting downward at a distance of 6.40 m up from the bottom of the ladder, the normal force (N) exerted by the ground acting perpendicular to the ground, and the frictional force (f) acting horizontally in the direction opposite to the potential sliding motion of the ladder.

Step 2: Write down the equations for force equilibrium.

In the vertical direction, the sum of the vertical forces must be zero:

N + Wf - WL = 0

In the horizontal direction, the sum of the horizontal forces must be zero:

f = 0 (since there is no horizontal acceleration)

Step 3: Express the forces in terms of known quantities.

WL = 350 N (given)

Wf = 885 N (given)

The angle of inclination of the ladder with respect to the ground is given as 54.0°.

Step 4: Calculate the normal force N.

Using the vertical force equilibrium equation, we can solve for N:

N = WL - Wf = 350 N - 885 N = -535 N (negative sign indicates that N acts in the opposite direction of WL and Wf)

Step 5: Calculate the frictional force f.

Since the ladder is on the verge of slipping, the frictional force f will be at its maximum value, which is given by:

f = μN, where μ is the coefficient of static friction.

Step 6: Calculate the coefficient of static friction μ.

Using the calculated value of N, we can now calculate μ:

μ = f / N = (-535 N) / N = -535

Step 7: Determine the minimum value of μ.

The coefficient of static friction cannot be negative, as it is always non-negative in reality. Therefore, the minimum value of the coefficient of static friction between the ladder and the ground is 0, which means that the ladder must have sufficient friction with the ground to prevent slipping.

In conclusion, the minimum value for the coefficient of static friction between the ladder and the ground, so that the ladder (with the firefighter on it) does not slip, is 0.