Concept explainers
A stepladder of negligible weight is constructed as shown in Figure P10.73, with AC = BC = ℓ = 4.00 m. A painter of mass m = 70.0 kg stands on the ladder d = 3.00 m from the bottom. Assuming the floor is frictionless, find (a) the tension in the horizontal bar DE connecting the two halves of the ladder, (b) the normal forces at A and B, and (c) the components of the reaction force at the single hinge C that the left half of the ladder exerts on the right half. Suggestion: Treat the ladder as a single object, but also treat each half of the ladder separately.
(a)
The tension experienced on the horizontal bar
Answer to Problem 73P
The tension experienced on the horizontal bar
Explanation of Solution
From the geometry of the ladder the angle
First of all consider the net torque at the point A which is at the bottom left side of the ladder. The torque experienced at this point is due to the weight of the painter, and the normal force at point B,
Write the expression for the torque at point A.
Here,
The net torque at point A is zero, hence equate equation (I) to zero and obtain an expression for
Now consider the net torque acting at the point B, which is at the bottom right of the ladder. The torque at point B same as A, is due to the weight of the painter and normal force at point A,
Write the expression for the net torque at point B.
The net torque acting at point B will be zero, equate equation (III) to zero, and obtain an expression for
Consider the torque at the point Cat the top of the right half of the ladder. At this point torque is due to the tension,
Write the expression for the torque at point C.
The net torque at point C is zero. Equate equation (III) to zero, and obtain an expression for
Substitute, equation (II) in (VI).
The tension
Write the expression for the force acting along right.
Equate equation (VIII) to zero.
Write the expression for the force acting along
Equate equation (VIII) to zero.
From equation (XI) the reaction force
Conclusion:
Substitute,
Therefore, the tension experienced on the horizontal bar
(b)
The normal force acting at A, and B.
Answer to Problem 73P
The normal force acting at A is
Explanation of Solution
Use equation (II) and (IV) to obtain the answer.
Conclusion:
Substitute,
Substitute,
Therefore, the normal force acting at A is
(c)
The components of reaction forces at point C.
Answer to Problem 73P
The components of reaction forces at point C are,
Explanation of Solution
Consider equation (IX) and (XI), the
The tension experienced on the horizontal bar
The normal force acting at B is
Conclusion:
Therefore, the components of reaction forces at point C are,
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Principles of Physics: A Calculus-Based Text
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