Three forces, $\overrightarrow { \mathbf { F } }$ ₁, $\overrightarrow { \mathbf { F } }$ ₂, and $\overrightarrow { \mathbf { F } }$ ₃, all act on an object, as shown in the figure. The magnitudes of the forces are: F₁ = 80.0 N, F₂ = 60.0 N, and F₃ = 40.0 N. The resultant force acting on the object is given by A) 180 N at an angle of 60.0° with respect to +x-axis. B) 60.0 N at an angle of 90.0° with respect to +x-axis. C) 20.0 N at an angle of 34.3° with respect to +x-axis. D) 35.5 N at an angle of 34.3° with respect to +x-axis. E) 40.0 N at an angle of 60.0° with respect to +x-axis.

Question

Quizplus · Accepted Answer

To find the resultant force, we can use the vector addition. We can first add $\overrightarrow { \mathbf { F } }$ ₁ and $\overrightarrow { \mathbf { F } }$ ₂ using the parallelogram method or the head-to-tail method. The magnitude of this resultant vector is: $$\left| \overrightarrow { \mathbf { R } } _ { 1 } \right|=\sqrt { \left( 80.0 \right) ^ { 2 }+\left( 60.0 \right) ^ { 2 }+2\left( 80.0 \right) \left( 60.0 \right) \cos { 120.0 }^{ \circ } } = 100.0\text{ N}$$ Next, we can add $\overrightarrow { \mathbf { R } } _ { 1 }$ and $\overrightarrow { \mathbf { F } }$ ₃ using the same method. The magnitude of the final resultant vector is: $$\left| \overrightarrow { \mathbf { R } } \right|=\sqrt { \left( 100.0 \right) ^ { 2 }+\left( 40.0 \right) ^ { 2 }+2\left( 100.0 \right) \left( 40.0 \right) \cos { 150.0 }^{ \circ } } = 70.8\text{ N}$$ Finally, we can find the direction of the resultant vector by taking the arctangent of the y-component divided by the x-component: $$\theta = \tan ^ { - 1 } \frac { \sum F _ { y } } { \sum F _ { x } } = \tan ^ { - 1 } \frac { 40.0 \sin { 150.0 }^{ \circ } } { 100.0 + 40.0 \cos { 150.0 }^{ \circ } } \approx 34.3^{ \circ }\text{ (with respect to the positive x-axis)}$$ Therefore, the answer is D.

Three Forces F→\overrightarrow { \mathbf { F } }F 1 F→\overrightarrow { \mathbf { F } }F

Three Forces $\overrightarrow { \mathbf { F } }$ ₁ $\overrightarrow { \mathbf { F } }$