$9$ . Three forces in a plane act on an object. The forces are $70$ pounds, $115$ pounds, and $135$ pounds. The $70$ pound force is exerted along the positive $x$ -axis. The $115$ pound force is applied below the $x$ -axis at a $120$ o angle with the $70$ pound force. The angle between the $115$ -pound and $135$ -pound forces is $75$ , and between the $135$ -pound and $70$ -pound forces is $165 \%$ . Are the vectors in equilibrium? If not, find the magnitude and the direction of the resultant force.

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9

. Three forces in a plane act on an object. The forces are

70

pounds,

115

pounds, and

135

pounds. The

70

pound force is exerted along the positive

x

-axis. The

115

pound force is applied below the

x

-axis at a

120

o angle with the

70

pound force. The angle between the

115

-pound and

135

-pound forces is

75

, and between the

135

-pound and

70

-pound forces is

165 \%

. Are the vectors in equilibrium? If not, find the magnitude and the direction of the resultant force.

Breakdown Forces: First, let's break down each force into its x and y components. The $70$ -pound force is along the positive x-axis, so its components are $(70, 0)$ .
Calculate Components $115$ -pound Force: The $115$ -pound force makes a $120$ -degree angle with the $70$ -pound force. To find its components, we use cosine for the x-component and sine for the y-component. The x-component is $115 \times \cos(120^\circ)$ and the y-component is $115 \times \sin(120^\circ)$ .
Calculate Components $135$ -pound Force: Calculating the components for the $115$ -pound force: $x$ -component = $115 \times \cos(120°) = -57.5$ pounds (since $\cos(120°) = -0.5$ ), $y$ -component = $115 \times \sin(120°) = 99.6$ pounds (since $\sin(120°) = 0.866$ ).
Add Components for Resultant Force: The $135$ -pound force makes a $75$ -degree angle with the $115$ -pound force, but we need the angle it makes with the positive x-axis to find its components. Since the angle between the $135$ -pound and $70$ -pound forces is $165^\circ$ , we use this angle for the components. The x-component is $135 \times \cos(165^\circ)$ and the y-component is $135 \times \sin(165^\circ)$ .
Calculate Magnitude Resultant Force: Calculating the components for the $135$ -pound force: $x$ -component $= 135 \times \cos(165°) = -130.9$ pounds (since $\cos(165°) = -0.966$ ), $y$ -component $= 135 \times \sin(165°) = 36.9$ pounds (since $\sin(165°) = 0.259$ ).
Find Direction Resultant Force: Now, we add up the x-components and y-components of all forces to find the resultant force components. Resultant x-component = $70 - 57.5 - 130.9$ . Resultant y-component = $0 + 99.6 + 36.9$ .
Find Direction Resultant Force: Now, we add up the $x$ -components and $y$ -components of all forces to find the resultant force components. Resultant $x$ -component $= 70 - 57.5 - 130.9$ . Resultant $y$ -component $= 0 + 99.6 + 36.9$ . Adding the components: Resultant $x$ -component $= -118.4$ pounds. Resultant $y$ -component $= 136.5$ pounds.
Find Direction Resultant Force: Now, we add up the x-components and y-components of all forces to find the resultant force components. Resultant x-component = $70 - 57.5 - 130.9$ . Resultant y-component = $0 + 99.6 + 36.9$ . Adding the components: Resultant x-component = $-118.4$ pounds. Resultant y-component = $136.5$ pounds. To find the magnitude of the resultant force, we use the Pythagorean theorem: magnitude = $\sqrt{(\text{x-component}^2 + \text{y-component}^2)}$ .
Find Direction Resultant Force: Now, we add up the x-components and y-components of all forces to find the resultant force components. Resultant x-component = $70 - 57.5 - 130.9$ . Resultant y-component = $0 + 99.6 + 36.9$ . Adding the components: Resultant x-component = $-118.4$ pounds. Resultant y-component = $136.5$ pounds. To find the magnitude of the resultant force, we use the Pythagorean theorem: magnitude = $\sqrt{(\text{x-component}^2 + \text{y-component}^2)}$ . Calculating the magnitude: magnitude = $\sqrt{(-118.4)^2 + (136.5)^2} = \sqrt{14019.36 + 18642.25} = \sqrt{32661.61} = 180.7$ pounds.
Find Direction Resultant Force: Now, we add up the x-components and y-components of all forces to find the resultant force components. Resultant x-component = $70 - 57.5 - 130.9$ . Resultant y-component = $0 + 99.6 + 36.9$ . Adding the components: Resultant x-component = $-118.4$ pounds. Resultant y-component = $136.5$ pounds. To find the magnitude of the resultant force, we use the Pythagorean theorem: magnitude = $\sqrt{(\text{x-component}^2 + \text{y-component}^2)}$ . Calculating the magnitude: magnitude = $\sqrt{(-118.4)^2 + (136.5)^2} = \sqrt{14019.36 + 18642.25} = \sqrt{32661.61} = 180.7$ pounds. To find the direction of the resultant force, we calculate the angle it makes with the positive x-axis using the tangent function: angle = $\arctan(\text{y-component} / \text{x-component})$ .
Find Direction Resultant Force: Now, we add up the x-components and y-components of all forces to find the resultant force components. Resultant x-component = $70 - 57.5 - 130.9$ . Resultant y-component = $0 + 99.6 + 36.9$ . Adding the components: Resultant x-component = $-118.4$ pounds. Resultant y-component = $136.5$ pounds. To find the magnitude of the resultant force, we use the Pythagorean theorem: magnitude = $\sqrt{(\text{x-component}^2 + \text{y-component}^2)}$ . Calculating the magnitude: magnitude = $\sqrt{(-118.4)^2 + (136.5)^2} = \sqrt{14019.36 + 18642.25} = \sqrt{32661.61} = 180.7$ pounds. To find the direction of the resultant force, we calculate the angle it makes with the positive x-axis using the tangent function: angle = $\arctan(\text{y-component} / \text{x-component})$ . Calculating the angle: angle = $\arctan(136.5 / -118.4)$ . This should be in the second quadrant because the x-component is negative and the y-component is positive.
Find Direction Resultant Force: Now, we add up the x-components and y-components of all forces to find the resultant force components. Resultant x-component = $70 - 57.5 - 130.9$ . Resultant y-component = $0 + 99.6 + 36.9$ . Adding the components: Resultant x-component = $-118.4$ pounds. Resultant y-component = $136.5$ pounds. To find the magnitude of the resultant force, we use the Pythagorean theorem: magnitude = $\sqrt{(\text{x-component})^2 + (\text{y-component})^2}$ . Calculating the magnitude: magnitude = $\sqrt{(-118.4)^2 + (136.5)^2} = \sqrt{14019.36 + 18642.25} = \sqrt{32661.61} = 180.7$ pounds. To find the direction of the resultant force, we calculate the angle it makes with the positive x-axis using the tangent function: angle = $\arctan(\text{y-component} / \text{x-component})$ . Calculating the angle: angle = $\arctan(136.5 / -118.4)$ . This should be in the second quadrant because the x-component is negative and the y-component is positive. The angle calculation is incorrect because the tangent of a negative divided by a negative is positive, which would place the angle in the first quadrant, not the second. The correct calculation should consider the signs of the components to determine the correct quadrant for the angle.