How Do You Calculate the Resultant Force and Its Direction from Two Vectors?

[shortyh90]

Homework Statement

Two ropes are attached to a heavy box to pull it along the floor. One rope applies a force of 470N in a direction due west; the other applies a force of 524 N in a direction due south. (A) How much force should be applied by a single rope and (B) in what direction (relative to due west) if it is to accomplish the same effect as the two forces being added together?

Homework Equations

The Attempt at a Solution

I drew the vectors and labeled them. Then I found the magnitude of the resultant by doing:

(a)SqRt (470)^2 + (524)^2
Answer: 495476
Then divided the answer by 2.

(b) tan(theta)=opp/adj
= 470/524
= 0.8969

tan^-1(0.8969)=41.9 degres

 

[JaredJames]

I will gladly help you with this question, however as per PF guidelines can you please show your attempt at this question first.

Jared

 

[shortyh90]

Thanks, the solution attempt has been posted.

 

[JaredJames]

OK, so for the magnitude of the force:

a^2 + b^2 = c^2

Square Root ( a^2 + b^2 ) = Square Root ( 470^2 + 524^2 ) = c

Where c is the magnitude of the resultant force. (In your attempt you forgot to root c and you don't divide by 2)

For the direction, once you have the magnitude using the above you then know hypotenuse and adjacent in relation to due west h = c (from above) and a = 470.

Using Soh Cah Toa:

Cos(theta) = (a/h), rearrange to give you the direction the rope needs to pull in relation to due west. Your solution of 41.9 degrees is the angle in relation to due south, not west.

Jared

 

[rwisz]

Therefore, the single rope would need to apply a force of approximately 495 N at an angle of 41.9 degrees south of due west in order to have the same effect as the two separate ropes. This can be achieved by using vector addition and finding the resultant force and direction. It is important to note that the direction of the single rope's force is not directly south, but rather at a slight angle, due to the combined effect of the two original forces. This highlights the importance of considering vector forces when solving problems involving multiple forces acting on an object.