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Solutions to problems related to computing the tensions in a cable given the weight and angles. It includes deriving expressions for the tensions using trigonometry and the addition formula for sine. The document also includes examples of computing the tensions for different weights and angles.
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Turn to the figure below. Assume that a weight W hangs from a cable as shown. The cable is anchored firmly on the left and right. The respective angles the cable makes with the horizontal are α 1 and α 2 respectively. The tensions in the cable—the tension in a cable
α α
1 2
is the magnitude with which the cable pulls—are T 1 and T 2 respectively.
Problem 1. Consider the weight W and the angles α 1 and α 2 as given and assume that the configuration depicted in the figure is stable. Draw a force diagram for the point at which the weight is suspended and use results of the section “Dealing with Forces” of Chapter 2 to express both T 1 and T 2 in terms of W and the angles α 1 and α 2. Conclude that if α 1 = α 2 , then T 1 = T 2.
Problem 2. Look up the addition formula for the sine and use it to simplify the expressions for T 1 and T 2 derived in Problem 1 to
T 1 = W cos α 2 sin(α 1 + α 2 ) and T 2 = W cos α 1 sin(α 1 + α 2 )
Problem 3. Assume that W = 500 pounds, α 1 = 10◦, and α 2 = 5◦^ and use your the formulas of Problem 2 to compute the tensions T 1 and T 2. Repeat your computation of T 1 and T 2 with W = 1000 pounds, α 1 = 5◦, and α 2 = 4◦. Finally, repeat the computations once more with W = 2000 pounds and the angles α 1 = 4◦^ and α 2 = 2◦.
Problem 4. The figure below is an abstraction of Image 6. It shows a cable pulling on a utility pole with a force of magnitude T at an angle β with the horizontal. Provide an
β
expression for the horizontal component of T. Let T = 20,000 pounds, take β equal to 40 ◦, 30 ◦, and 20◦, and compute the magnitude of this horizontal component in each case. Do you think that this force is problematic in the context of the solution of the structural problem?
About Steel Cables. A 12 -inch diameter steel cable typically has a minimum breaking strength of about 20,000 to 25,000 pounds and can support a load between 4000 and 5000 pounds safely. For a 34 -inch diameter steel cable, these ranges are from about 45,000 to 50,000 pounds and 9,000 to 10,000 pounds.
About Utility Poles. The force that a vertical wooden pole can support depends on a number of factors including the type of wood, the thickness of the pole, the depth and quality of the foundation, and the height and direction at which the force is applied. The figure and table below provide very general guidelines (for applications in the U.S.).
Minimum Diameter of Pole Length Range of Pole Allowable Horizontal Load (in inches) (in feet) (in pounds) 12.4 45-125 11, 11.8 45-125 10, 11.1 40-125 8, 10.5 40-125 7, 9.9 35-125 6,