For the stepladder shown in Fig. 12-51, sides AC and CE are each 3.34 m long and hinged at C. Bar BD is a tie-rod 0.662 m long, halfway up. A man weighing 746 N climbs 2.51 m along the ladder. Assuming that the floor is frictionless and neglecting the mass of the ladder, find (a) the tension in the tie-rod and the magnitudes of the forces on the ladder from the floor at (b) A and (c) E. (Hint: Isolate parts of the ladder in applying the equilibrium conditions.)
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To solve this problem, we can use the laws of equilibrium and divide the ladder into two parts: the left side and the right side.
First, let's find the tension in the tie-rod:
Since the tie-rod is halfway up the ladder, we can consider the left and right sides as separate free-body diagrams.
Left side:
The forces acting on the left side of the ladder are:
- The weight of the man (746 N) acting downward at point D.
- The tension in the tie-rod acting upward at point D.
- The natural reaction force exerted by the floor at point A.
Right side:
The forces acting on the right side of the ladder are symmetrical to those on the left side.
- The weight of the man (746 N) acting downward at point D.
- The tension in the tie-rod acting upward at point D.
- The natural reaction force exerted by the floor at point E.
Using the laws of equilibrium, the sum of vertical forces and the sum of horizontal forces for both sides will be zero.
(a) Tension in the tie-rod:
Since the ladder is in equilibrium, the weight of the man will be balanced by the tension in the tie-rod. Therefore, the tension in the tie-rod will be equal to the weight of the man.
Tension in the tie-rod = Weight of the man = 746 N.
(b) Force on the ladder from the floor at point A:
For the left side of the ladder, the sum of vertical forces is given by:
Tension in the tie-rod - Force at A = 0.
Therefore, the force on the ladder from the floor at point A is equal to the tension in the tie-rod.
Force at A = Tension in the tie-rod = 746 N.
(c) Force on the ladder from the floor at point E:
For the right side of the ladder, the sum of vertical forces is given by:
Tension in the tie-rod - Force at E = 0.
Therefore, the force on the ladder from the floor at point E is equal to the tension in the tie-rod.
Force at E = Tension in the tie-rod = 746 N.
So, the tension in the tie-rod is 746 N, and the magnitude of the forces on the ladder from the floor at points A and E is also 746 N.
To find the tension in the tie-rod and the magnitudes of the forces on the ladder from the floor at points A and E, we can use the principles of static equilibrium.
Let's start by isolating the upper part of the ladder ACDE. This part of the ladder is in equilibrium, meaning the sum of the forces acting on it is zero.
Considering the vertical forces, the weight of the man (746 N) acts downwards, and the force at point E from the floor acts upward. These two forces must balance each other, so we have:
746 N - F(floor at E) = 0
Therefore, F(floor at E) = 746 N.
Next, let's consider the horizontal forces on the upper part of the ladder. Since there is no horizontal acceleration, the sum of the horizontal forces must also be zero.
The only horizontal force acting on this part of the ladder is the tension in the tie-rod. So we get:
Tension in tie-rod - F(floor at A) = 0
Therefore, F(floor at A) = Tension in the tie-rod.
To find the tension in the tie-rod, we can consider the torque about point C. Since the ladder is in equilibrium, the net torque about any point must be zero.
We have the weight of the man as a clockwise torque and the tension in the tie-rod as a counterclockwise torque. The torque is given by the product of the force and the perpendicular distance from the point of rotation. In this case, the perpendicular distance is half of the tie-rod length, i.e., 0.662/2 = 0.331 m.
Therefore, we can write:
Torque(clockwise) = Torque(counterclockwise)
(746 N) * (2.51 m) = (Tension in tie-rod) * (0.331 m)
Solving for the tension in the tie-rod, we find:
Tension in tie-rod = (746 N * 2.51 m) / (0.331 m)
Now you can calculate the tension in the tie-rod using the given values.
To summarize:
(a) The tension in the tie-rod can be found using the equation: Tension in tie-rod = (746 N * 2.51 m) / (0.331 m)
(b) The force on the ladder from the floor at point A is equal to the tension in the tie-rod.
(c) The force on the ladder from the floor at point E is equal to the weight of the man (746 N).