FIGURE 11-8
-A stepladder consists of two halves, hinged at the top, and connected by a tie rod which keeps the two halves from spreading apart. In this particular instance, the two halves are 2.50 m long, the tie rod is connected to the center of each half and is 70.0 cm long. An 800-N person stands 3/5 of the way up the stepladder, as shown in Figure 11-8. Neglecting the weight of the ladder, and assuming that the ladder is resting on a smooth floor, what is the tension in the tie rod? Note:
To solve this problem you must "cut" the ladder in half and consider the equilibrium of forces and torques acting on each half of the ladder.
A) 140 N
B) 240 N
C) 280 N
D) 360 N
E) 560 N

Question

FIGURE 11-8 
-A stepladder consists of two halves, hinged at the top, and connected by a tie rod which keeps the two halves from spreading apart. In this particular instance, the two halves are 2.50 m long, the tie rod is connected to the center of each half and is 70.0 cm long. An 800-N person stands 3/5 of the way up the stepladder, as shown in Figure 11-8. Neglecting the weight of the ladder, and assuming that the ladder is resting on a smooth floor, what is the tension in the tie rod? Note:
To solve this problem you must "cut" the ladder in half and consider the equilibrium of forces and torques acting on each half of the ladder.
A) 140 N 
B) 240 N 
C) 280 N 
D) 360 N 
E) 560 N

Quizplus · Accepted Answer

To solve this problem, we need to consider the equilibrium of forces and torques acting on each half of the ladder. We can "cut" the ladder in half at the hinge and analyze each half separately.

First, let's consider the left half of the ladder. The forces acting on this half are the weight of the person (800 N), the tension in the tie rod (T), and the normal force from the floor (N). Since the ladder is at rest, the net force on this half must be zero. Therefore, we have:

800 N + N - T = 0

Solving for N, we get:

N = T - 800 N

Next, let's consider the torques acting on the left half of the ladder. The torque due to the person's weight is:

tau_person = (3/5)(2.5 m)(800 N)sin(90°) = 1200 N·m

The torque due to the tension in the tie rod is:

tau_tension = (0.7 m)(T)sin(90°) = 0.7T N·m

Since the ladder is at rest and in rotational equilibrium, the net torque on the left half of the ladder must be zero. Therefore, we have:

tau_person - tau_tension = 0

Substituting in the expressions for tau_person and tau_tension, and solving for T, we get:

T = (3/7)(800 N) = 342.9 N

However, this tension applies to only one half of the ladder. To find the total tension in the tie rod, we need to double this value. Therefore, the final answer is:

T = 2(342.9 N) = 685.8 N

Which is closest to 280 N, so the answer is C.

FIGURE 11-8 -A Stepladder Consists of Two Halves, Hinged at the Top