Question

Two forces with magnitudes of 150 and 75 pounds act on an object at angles of 30° and 150° respectively. Find the direction and magnitude of the resultant force. Round to two decimal places in all intermediate steps and in your final answer.

Answer 1

F_x = 150cos 30 - 75cos 30 = 64.95
F_y = 150sin 30 + 75sin 30 = 112.5

R = sqrt(64.95^2 + 112.5^2) = sqrt(16874.75) = 129.90
Direction of R is arctan(112.5/64.95) = arctan(1.73) = 59.97°

Answer 2

Magnitude= 129.55N

Direction= 60°

Explanation:

In order to find the magnitude, we must find the magnitude in both vertical and horizontal direction.

In horizontal direction:
`f_(x)`=150cos30°+ 75cos150°

=
`150`×
`(√(3) )/(2)` +
`75`×
`-(√(3) )/(2)`

=
`129.90-64.95`

=
`64.95N`

In vertical direction:
`f_(y)`= 150sin30°+75sin150°

=
`150`×
`(1)/(2)`+
`75`×
`(1)/(2)`

=
`112.50N`

Thus, magnitude is given by:
`\sqrt{f_(x) ^(2)+f_(y) ^(2) }`

=
`\sqrt{64.25^(2)+112.50^(2)  }`

=
`√(16784.31)`

=
`129.55N`

Direction of the force is calculated as: tanα=
`(f_(y) )/(f_(x) )`

=
`(112.50)/(64.95)`

=
`1.7321`

α=60°