Explanation Given data: A person is suspended by ropes tied to two trees. One rope makes an angle of 45 ° with the ground and the other makes an angle of 30 ° . Consider the rope that makes an angle of 30 ° with the ground is tied to the left side tree and the rope that makes an angle of 45 ° with the ground is tied to the right side tree. Weight of the person ( w ) is 560 N. Formula used: Write the expression for equilibrium equation of tension force for the x -component as follows. ∑ F x = 0 (1) Here, ∑ F x is the sum of the forces in the x -component. Write the expression for equilibrium equation of tension force for the y -component as follows. ∑ F y = 0 (2) Here, ∑ F y is the sum of the forces in the y -component. Explanation: From the given data, draw the figure as shown in Figure 1. From Figure 1, writ the values of angles made by the rope with the ground (horizontal) as follows. θ left = 30 ° θ right = 45 ° From the Newton’s first law, the sum of the forces acts on the riders towards each component ( x -component and y -component) is zero. Modify the expression in equation (1) for the forces in the x -component as follows. ( T left ) x + ( T right ) x = 0 (3) Here, ( T left ) x is the tension force T left (tension force in the left side rope) in the x -component, and ( T right ) x is the tension force T right (tension force in the right side rope) in the x -component. From the given figure, write the tension force ( T left ) x as follows. ( T left ) x = − T left cos ( θ left ) From the figure, write the tension force ( T right ) x as follows. ( T right ) x = T right cos ( θ right ) Substitute [ − T left cos ( θ left ) ] for ( T left ) x , and T right cos ( θ right ) for ( T right ) x in equation (4), [ − T left cos ( θ left ) ] + [ T right cos ( θ right ) ] = 0 T left cos ( θ left ) = T right cos ( θ right ) Write the expression for tension in the right side rope as follows. T right = T left [ cos ( θ left ) cos ( θ right ) ] Substitute 30 ° for θ left , and 45 ° for θ right , T right = T left [ cos ( 30 ° ) cos ( 45 ° ) ] = T left ( 0.8660 0.7071 ) T right = 1.2247 T left (4) Modify the expression in equation (2) for the forces in the y -component as follows. ( T left ) y + ( T right ) y − w = 0 (5) Here, ( T left ) y is the tension force T left in the y -component, ( T right ) y is the tension force T right in the y -component, and w is the weight of the person

4th Edition

ISBN: 8220106755235

Author: Field

Publisher: PEARSON

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Question

Chapter 5, Problem 7P

To determine

To find: The tension in each of the ropes.

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