Concept explainers
The 10-ft cable AB is attached to two collars as shown. The collar at A can slide freely along the rod; a stop attached to the rod prevents the collar at B from moving on the rod. Neglecting the effect of friction and the weight of the collars, determine the distance a.
Fig. P7.147
Find the distance a.
Answer to Problem 7.147P
The distance a is
Explanation of Solution
Given information:
The length of the cable AB is
The value of angle
The collar at A is slides freely and the collar at B is prevented from the moving.
Calculation:
Show the free-body diagram of the cable assembly as in Figure 1.
Refer the Equation 7.16 in the textbook.
Write the equation of the catenary cable as follows;
Differentiate the equation with x;
The slope at point A is;
The length of the portion AC is as follows:
The length of the portion CB is as follows:
Find the distance
Substitute 10 ft for L,
Find the distance
Find the distance
Consider the triangle ABD;
Find the value of
Find the distance a using the relation.
Use the trial and error procedure to find the value of a.
Consider the value of c and for the given value of
Trial 1:
Consider a trial value of 1.60 ft for c.
Substitute 1.60 ft for c and
Substitute 1.60 ft for c and 2.107 ft for
Substitute 1.60 ft for c and 2.107 ft for
Substitute 1.60 ft for c and 3.541 ft for
Substitute 2.107 ft for
The calculated value of
Trial 2:
Consider a trial value of 1.70 ft for c.
Substitute 1.70 ft for c and
Substitute 1.70 ft for c and 2.239 ft for
Substitute 1.70 ft for c and 2.239 ft for
Substitute 1.70 ft for c and 3.622 ft for
Substitute 2.239 ft for
The calculated value of
Trial 3:
Consider a trial value of 1.803 ft for c.
Substitute 1.803 ft for c and
Substitute 1.803 ft for c and 2.374 ft for
Substitute 1.803 ft for c and 2.374 ft for
Substitute 1.803 ft for c and 3.694 ft for
Substitute 2.374 ft for
The calculated value of
Therefore, the value of c is 1.803 ft.
Substitute 3.606 ft for
Therefore, the distance a is
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Chapter 7 Solutions
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