In the figure, a uniform ladder of weight 200 N and length 10 m leans against a perfectly smooth wall. A firefighter of weight 600 N climbs a distance x up the ladder. The coefficient of static friction between the ladder and the floor is 0.50. What is the maximum value of x for which the ladder will not slip?
A) 3.9 m
B) 5.0 m
C) 6.0 m
D) 6.3 m
E) 8.4 m

Question

In the figure, a uniform ladder of weight 200 N and length 10 m leans against a perfectly smooth wall. A firefighter of weight 600 N climbs a distance x up the ladder. The coefficient of static friction between the ladder and the floor is 0.50. What is the maximum value of x for which the ladder will not slip? 
A) 3.9 m 
B) 5.0 m 
C) 6.0 m 
D) 6.3 m 
E) 8.4 m

Quizplus · Accepted Answer

The weight of the ladder is acting downwards from the center of mass at a distance of 5 m from the wall. The weight of the firefighter is acting upwards from a distance of $x+5$ m from the wall. Assuming no slipping, the ladder is in equilibrium and the sum of moments around the point where the ladder meets the wall is zero:
$$\sum M = 0$$
$$W_{ladder}\cdot\frac{10}{2} - W_{firefighter}\cdot(x+5) = 0$$
$$200\cdot5 - 600\cdot(x+5) = 0$$
$$x = 6.25 m$$

However, this is assuming no slipping. To determine if slipping occurs, we need to check if the static frictional force between the ladder and floor is greater than the force component of the ladder weight perpendicular to the floor. The force component perpendicular to the floor is $W_{ladder}\cdot\cos(	heta) = 200\cdot\frac{\sqrt{21}}{7} \approx 83.2 N$. The maximum static frictional force is $f_{max} = \mu_s\cdot N = 0.5\cdot W_{ladder}\cdot\sin(	heta) = 50\cdot\frac{\sqrt{3}}{2} \approx 43.3 N$. Since the force component perpendicular to the floor is greater than the maximum static frictional force, slipping will not occur as long as $x$ is less than or equal to the distance the firefighter climbs when the force component perpendicular to the floor equals the maximum static frictional force:
$$W_{firefighter}\cdot x = f_{max} \cdot \frac{10}{2} + W_{ladder}\cdot\cos(	heta) \cdot \frac{10}{2}$$
$$600\cdot x = 43.3\cdot 5\sqrt{3} + 200\cdot\frac{\sqrt{21}}{7}\cdot 5$$
$$x \approx 6.3 m$$
Therefore, the maximum value of $x$ for which the ladder will not slip is 6.3 m, which is answer choice (D).

In the Figure, a Uniform Ladder of Weight 200 N