Precalculus with Limits: A Graphing Approach
Precalculus with Limits: A Graphing Approach
7th Edition
ISBN: 9781305071711
Author: Ron Larson
Publisher: Brooks/Cole
Question
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Chapter 10.2, Problem 74E
To determine

To find: The tension in each of the supporting cables if the weight of a crate is 500 newtons.

Expert Solution & Answer
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Answer to Problem 74E

The tension in each of the supporting cables are 241.14 N , 185.6 N and 266.80 N respectively.

Explanation of Solution

Given information: A crate of weight 500 newtons is supported by three cables, as shown in the figure below.

  Precalculus with Limits: A Graphing Approach, Chapter 10.2, Problem 74E

Figure (1)

Calculation:

Three upward forces AB , AC and AD are acting on the load. The forces are in the form vector. The coordinates of points A is (0, 0, −115) , B is (0, 70, 0) , C is (−60, 0, 0) and D is (45, −65, 0) .

The vector AB is,

  AB=〈0−0, 70−0, 0−(−115)〉=〈0, 70, 115〉

The vector AC is,

  AC=〈−60−0, 0−0, 0−(−115)〉=〈−60, 0, 115〉

The vector AD is,

  AC=〈45−0, −65−0, 0−(−115)〉=〈45, −65, 115〉

One downward force is also acting on the load that is the weight 500 newton. Let w be the force and its vector form is:

  w=〈0,0,−500〉

The unit vector in direction of AB is AB→‖AB→‖ . Let a be the force in direction of AB can be calculated as:

  a=‖a‖AB→‖AB→‖=‖a‖〈0,70,115〉0+702+1152=‖a‖〈0,70,115〉2529=‖a‖〈0,0.52, 0.48〉

The unit vector in direction of AC is AC→‖AC→‖ . Let b be the force in direction of AC can be calculated as:

  b=‖b‖AC→‖AC→‖=‖b‖〈−60,0,115〉3600+13225=‖b‖〈−60,0,115〉5673=‖b‖〈−0.46, 0, 0.89〉

The unit vector in direction of AD is AD→‖AD→‖ . Let c be the force in direction of AD can be calculated as:

  c=‖c‖AD→‖AD→‖=‖c‖〈45,−65,115〉452+(−65)2+1152=‖c‖〈45,−65,115〉779=‖c‖〈0.32,−0.47,0.82〉

At equilibrium, the resultant action on load is zero. Therefore

  a+b+c−w=0

The equations formed are:

  −0.46‖b‖+0.32‖c‖=00.52‖a‖−0.47‖c‖=00.48‖a‖+0.89‖b‖−0.82‖c‖=500

Solve the above equations.

  ‖c‖=266.80 N‖b‖=185.6 N‖a‖=241.14 N

Therefore, the tension in each of the supporting cables are 241.14 N , 185.6 N and 266.80 N respectively.

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Chapter 10.2, Problem 73E
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Precalculus with Limits: A Graphing Approach

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