Physics Help?

Question

Consdier four vectors with magnitudes of f1 = 49 N , f2 = 32N, f3= 28 N, f4= 41N. Let θ1 = 120 , θ2 = -150, θ3=31 , and θ4= -63, measured form the positive x axis with the counter clockwise angulare direction as postive.

What is the magnitude of the reultant vector F , where F=f1+f2+f3+f4. ANswer in units of N.

What i did was i took the sin θ of each and multiplyed by the magnitudes and added them im not sure if that is right but i got answer of 4.325043791.

If that is the right answer how do i get direction
What is the direction of this resultant F. Give the angle in degrees, use counterclockwise as the postive angular direction, between the limits of -180 and 180 from the postive x axis.

Feb 23 2007 12:17 pm -47.6439
Feb 23 2007 12:19 pm -24.2559
Feb 23 2007 12:30 pm -65.74
Feb 23 2007 12:30 pm 294.26
Feb 23 2007 12:41 pm 335.744
Those are all wrong answers for the angle

Answer 1

You first break each vector into components:

F1=49cos(120)i+49sin(120)j
F2=32cos(-150)i+32sin(-150)j
F3=28cos(31)i+28sin(31)j
F4=41cos(-63)i+41sin(-63)j

F_total=F1+F2+F3+F4
F_total=[49cos(120)+32cos(-150)+28cos(31)+41cos(-63)]i + [49sin(120)+32sin(-150)+28sin(31)+41sin(-63)]j
F_total=-9.60i+4.32j

The magnitude of F_total you can find by square root of the sum of squares of the two component lengths. Note from the signs of the two components that F_total is in the second quadrant. According to yout definition for angles, this is going to give you a theta between 90 and 180 degrees. Now first find the magnitude of the resultant force:

F_total=10.53N

To find the angle, you can use arcsin or arccos. Try them both.

theta1=arcsin(4.32/10.53)=24.25 degrees
theta2=arccos(-9.6/10.53)=155.74 degrees

Since you already know that the resultant force is in the second quadrant, the answer is theta2=155.74 degrees. Note that theta1 is not exactly wrong. Because the sin (theta)=sin(180-theta), the function arcsin will typically give you the just the lowest number. This isn't necessarily what you want in this case. You need to be able to discriminate and transform, which is why it helps to know which quadrant the resultant is in based on the components. This will help you immensely in finding the right angles when working with arccos and arcsin to find resultant vector angles in the definition of angles that you want.

Answer 2

The easiest way to answer this question is graphically. Using lined paper, you draw each vector to scale and to the angles given. So, you start with F1 and draw a line 49mm at the 120deg angle. From the end point of that, draw F2 at a length of32mm at an angle of 150, and so on. After you've added all the vectors head to tail, draw a line for the resultant, measure it's length and angle and you're done.

The other, more precise way to do it is to break each force into x and y components, and summing all of the x's to get the resultant x and summing all of the y's to get the resultant y. The magnitude of the resultant is the square root of the sum of the resultants squared (Pythagorean theorm), and the angle is the arctan of the resultant y divided by the resultant x. I'll get you started...

F1x = -49*cos 60 (it's - because it is pointing to the left)
F1y = 49*sin 60
F2x = -32*cos 30 (same as F1x)
F2y = 32*sin 30
F3x = 28*cos 31
F3y = 28*sin 31
F4x = 41*cos 63
F4y = 41*sin 63

Rx = -F1x-F2x+F3x+F4x
Ry = F1y+F2y+F3y+F4y

R = sqrt (Rx^2 + Ry^2)
angle = arctan (Ry/Rx)

Answer 3

First you have to find the resultant in the y-direction, which is what you started doing

F*sin(angle) =F(y)

do that for the 4 forces and add them up

F(y)=49*sin(120)+32*sin(-150)+ 28*sin(31)+41*sin(-63)
F(y)=4.3250

That is the same thing you got, now you do this same thing except with cos to get resultant force in x direction F(x)

F(x) =49*cos(120)+32*cos(-150) +28*cos(31)+41*cos(-63)
F(x)= -9.5985

Now this is like the base and height of a triangle and you use pythagreom thm to find the result magnitude

sqrt(F)=F(x)^2 + F(y)^2
F= 10.5279

To find the angle for this F, use this formula:

tan(angle) = F(y)/F(x)
tan(angle) = -2.2193

use the inverse tan button to find the angle

angle = -65.74 degrees from your positive x

F = 10.5279 at - 65.74 degrees

Answer 4

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