Question

Three forces with magnitudes of 66 pounds, 110 pounds, and 138 pounds act on an object at angles of 30°, 45°, and 120° respectively, with the positive x-axis. Find the direction and magnitude of the resultant force. (Round your answers to one decimal place.)

Answer 1

• magnitude = 239.5 lbf
• direction ( angle to the x axis) = 74.0

Step-by-step explanation:

We just need to sum the forces, we can do this easily in their Cartesian form.

Knowing the magnitude and angle with the positive x axis, we can find Cartesian representation of the vectors using the formula

`\vec{A}= |\vec{A}| \ ( \ cos(\theta) \ , \ sin (\theta) \ )`

where
`|\vec{A}|` its the magnitude of the vector and θ the angle with the positive x axis.

So, for our forces we got:

`\vec{F}_1 \ = \ 66 \ lbf \ * \ ( \ cos (30\°)\ , \ sin(30\°) \ )`

`\vec{F}_2 \ = \ 110 \ lbf \ * \ ( \ cos (45\°)\ , \ sin(45\°) \ )`

`\vec{F}_3 \ = \ 138 \ lbf \ * \ ( \ cos (120\°)\ , \ sin(120\°) \ )`

this will give us:

`\vec{F}_1 \ = ( \ 57.157 \ lbf \ , \ 33 \ lbf \ )`

`\vec{F}_2 \ = ( \ 77.782 \ lbf \ , \ 77.782 \ lbf \ )`

`\vec{F}_3 \ = ( \ - 69 \ lbf \ , \ 119.511 \ lbf \ )`

Now, we just sum the forces:

`\vec{F}_(net) \ = \ \vec{F}_1 \ + \ \vec{F}_2 \ + \ \vec{F}_3`

`\vec{F}_(net) \ = ( \ 57.157 \ lbf \ , \ 33 \ lbf \ ) + ( \ 77.782 \ lbf \ , \ 77.782 \ lbf \ ) + (\ - 69 \ lbf \ , \ 119.511 \ lbf \ )`

`\vec{F}_(net) \ = ( \ 57.157 \ lbf \ + \ 77.782 \ lbf \ - \ 69 \ lbf \, \ 33 \ lbf \ + \ 77.782 \ lbf \ + \ 119.511 \ lbf \ )`

`\vec{F}_(net) \ = ( \ 65.939 \ lbf \, \ 230.293 lbf \ )`

This is the net force, to obtain the magnitude, we just need to find the length of the vector, using the Pythagorean formula:

`|\vec{F}_(net)| = \sqrt{(F_(net_x))^2+(F_(net_y))^2}`

`|\vec{F}_(net)| = √((65.939 \ lbf)^2+(230.293 lbf)^2)`

`|\vec{F}_(net)| = \ 239.547 \ lbf`

To obtain the angle with the positive x-axis we can use the formula:

`\theta \ = \ arctan( (F_y)/(F_y))`

`\theta \ = \ arctan( (230.293 lbf)/(65.939 \ lbf))`

`\theta \ = \ arctan( 3.492)`

`\theta \ = \ 74.02`

So, the answer its

`magnitude = \ 239.547 \ lbf`

`angle_( (to the x axis)) = \ 74.02`

Rounding up:

`magnitude = \ 239.5 \ lbf`

`angle_( (to the x axis)) = \ 74.0`