Three forces with magnitudes pounds, pounds and pounds act on an object at angles , , and respectively, with the positive x-axis. Find the direction and magnitude of the resultant force. (The choices below are given to two decimal places.)
A) with the positive x-axis
B) with the positive x-axis
C) with the positive x-axis
D) with the positive x-axis
E) pounds at an angle of with the positive x-axis

Question

Three forces with magnitudes  pounds,  pounds and  pounds act on an object at angles  ,  , and  respectively, with the positive x-axis. Find the direction and magnitude of the resultant force. (The choices below are given to two decimal places.)
A) with the positive x-axis 
B) with the positive x-axis 
C) with the positive x-axis 
D) with the positive x-axis 
E) pounds at an angle of with the positive x-axis

Quizplus · Accepted Answer

To find the resultant force, we need to add the x and y components of the individual forces. Using the given angles and magnitudes, we can find the x and y components as follows:

Force 1:
Magnitude = 11eaa595_6408_cdda_a696_65d482aea31c_TB4584_11_TB4584_11 pounds
Direction = 11eaa595_6409_1bfd_a696_f17a2f4c9bce_TB4584_11_TB4584_11
x-component = Magnitude*cos(Direction) = 11eaa595_6409_2c0a_a696_a9a93f26bc21_TB4584_11_TB4584_11 pounds
y-component = Magnitude*sin(Direction) = -11eaa595_6409_2c0a_a696_a9a93f26bc21_TB4584_11_TB4584_11 pounds (negative because it is in the negative y-axis direction)

Force 2:
Magnitude = 11eaa595_6408_f4eb_a696_efbf02a9aa96_TB4584_11_TB4584_11 pounds
Direction = 11eaa595_6409_430e_a696_8bc58923c93e_TB4584_11_TB4584_11
x-component = Magnitude*cos(Direction) = -11eaa595_6409_6a1c_a696_64e8ca0946f1_TB4584_11_TB4584_11 pounds (negative because it is in the negative x-axis direction)
y-component = Magnitude*sin(Direction) = 11eaa595_6409_6a1c_a696_64e8ca0946f1_TB4584_11_TB4584_11 pounds

Force 3:
Magnitude = 11eaa595_6409_1bfc_a696_b73e4c6f90eb_TB4584_11_TB4584_11 pounds
Direction = 11eaa595_6409_430f_a696_7f84f4f9d456_TB4584_11_TB4584_11
x-component = Magnitude*cos(Direction) = 11eaa595_6409_6a20_a696_7317147db258_TB4584_11_TB4584_11 pounds
y-component = Magnitude*sin(Direction) = -11eaa595_6409_6a20_a696_7317147db258_TB4584_11_TB4584_11 pounds (negative because it is in the negative y-axis direction)

To find the x and y components of the resultant force, we add the x and y components of the individual forces:

Resultant x-component = (11eaa595_6409_2c0a_a696_a9a93f26bc21_TB4584_11_TB4584_11 - 11eaa595_6409_6a1c_a696_64e8ca0946f1_TB4584_11_TB4584_11 + 11eaa595_6409_6a20_a696_7317147db258_TB4584_11_TB4584_11) pounds = 11eaa595_6409_b844_a696_2de055b8dbc1_TB4584_11_TB4584_11 pounds

Resultant y-component = (-11eaa595_6409_2c0a_a696_a9a93f26bc21_TB4584_11_TB4584_11 + 11eaa595_6409_6a1c_a696_64e8ca0946f1_TB4584_11_TB4584_11 - 11eaa595_6409_6a20_a696_7317147db258_TB4584_11_TB4584_11) pounds = -11eaa595_6409_b844_a696_2de055b8dbc1_TB4584_11_TB4584_11 pounds (negative because it is in the negative y-axis direction)

To find the magnitude and direction of the resultant force, we use the Pythagorean theorem and the inverse tangent function:

Magnitude = sqrt(Resultant x-component^2 + Resultant y-component^2) = 11eaa595_6409_b843_a696_051e5895499e_TB4584_11_TB4584_11 pounds

Direction = atan(Resultant y-component/Resultant x-component) = atan(-1) = 11eaa595_6409_df55_a696_7950fa6db278_TB4584_11_TB4584_11 with the positive x-axis

Therefore, the answer is E, 11eaa595_6409_b844_a696_2de055b8dbc1_TB4584_11_TB4584_11 pounds at an angle of 11eaa595_6409_df55_a696_7950fa6db278_TB4584_11_TB4584_11 with the positive x-axis.

Three Forces with Magnitudes Pounds, Pounds and