A ladder can fall for two reasons. If it is set too steep and climber gets their mass to the left of the ladder's base, the ladder likely will fall over backwards. If the ladder is set at too shallow of an angle the required force of friction between the ladder and the ground might be too great and the base of the ladder will slip. Assume that there is no friction between the ladder and the wall and that the ladder is effectively weightless. The coefficient of friction between the base of the ladder and ground is 0.29. The person using the ladder will be 3/4 of the way up the ladder. If the person climbing the ladder has a weight of 980 newtons and the ladder is 3.97 meters long, how far from the wall can the base of the ladder be placed, to the nearest hundredth of a meter, and not slip?

To determine how far from the wall the base of the ladder can be placed without slipping, we need to consider the forces acting on the ladder.

First, let's address the slipping condition. The maximum force of friction required to prevent slipping can be calculated by multiplying the coefficient of friction (µ) with the normal force (N). The normal force can be determined by considering the forces acting on the climber.

The weight of the climber (980 newtons) can be divided into two components: one parallel to the ladder and one perpendicular to the ladder. The component parallel to the ladder will be equal to the weight multiplied by the cosine of the angle of the ladder with respect to the ground.

The perpendicular component of the weight will be equal to the weight multiplied by the sine of the angle of the ladder. This component is responsible for applying the normal force to the ladder, which determines the force of friction against slipping.

Since the person is located 3/4 of the way up the ladder, we can determine the angle of the ladder by using the inverse sine function. The angle (θ) can be calculated as:

θ = arcsin(3/4)

Next, we can calculate the maximum force of friction (Ffriction) using the coefficient of friction (µ) and the normal force (N):

Ffriction = µ * N

Finally, we can set up an equation to determine the maximum distance from the wall (x) at which the ladder will not slip:

Ffriction = (weight parallel to ladder) = µ * N

(weight parallel to ladder) = weight * cos(θ)

Since the ladder is effectively weightless, we can substitute the weight with 0 in the equation:

µ * N = 0 * cos(θ)

By rearranging the equation and substituting in the known values, we can solve for x:

x = (weight perpendicular to ladder) / (µ * g * cos(θ))

(weight perpendicular to ladder) = weight * sin(θ)

Substituting the values, we can calculate the maximum distance from the wall:

x = (weight * sin(θ)) / (µ * g * cos(θ))

Now, plugging in the given values:
- weight (W) = 980 newtons
- coefficient of friction (µ) = 0.29
- g = acceleration due to gravity = 9.8 m/s^2

θ = arcsin(3/4)
θ ≈ 48.59 degrees

x = (980 * sin(48.59)) / (0.29 * 9.8 * cos(48.59))

Evaluating this expression, we find:

x ≈ 2.83 meters

Therefore, the base of the ladder can be placed up to approximately 2.83 meters from the wall without slipping.