A force of 200n is inclined at an angle of 120° to another force P. The angle between the 200n force and the resultant force is 50°. Find the magnitude of the force P and the resultant of the two force.?

Answers:  

Magnitude of force P = 163.041494

Magnitude of resultant force = 184.320997

Values are approximate. Units are in newtons.

Explanation:

Let for P be pulled directly to the east. The vector for this force is <x, 0> where x is positive.

The 200 newton force has the vector <200*cos(120), 200*sin(120)>

The resultant vector is <x+200*cos(120),200*sin(120)>. Each component is the sum of the corresponding components of <x,0> and <200*cos(120), 200*sin(120)>

The resultant vector is also <r*cos(70),r*sin(70)>. Note how 70+50 = 120. The 50 degree angle is known, so we effectively do 120-50 = 70 to find the angle of the resultant vector with the positive x axis.

The resultant vector expressions we found were

Equate the y components of each resultant vector expression. Solve for r

r*sin(70) = 200*sin(120)

r = 200*sin(120)/sin(70)

r = 184.320997021376

Make sure your calculator is in degree mode.

Let's round this r value to 6 decimal places to simplify things a bit

r = 184.320997

Now equate the x components of each resultant vector expression, plug in the r value we found, and solve for x

x+200*cos(120) = r*cos(70)

x+200*cos(120) = 184.320997*cos(70)

x = 184.320997*cos(70) - 200*cos(120)

x = 163.041494

this value is approximate just like r is as well

The magnitude of force P is

magnitude = sqrt(x^2+y^2)

magnitude = sqrt(x^2+0^2)

magnitude = sqrt((163.041494)^2+0^2)

magnitude = 163.041494

Which is equal to the x value. This applies because y = 0.

The magnitude of the resultant is r = 184.320997 which we found earlier

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call the larger triangle abc

then the smaller triangle cde is similar to abc (all angles are equal, aaa).  

ab is the side that is 36, and we wish to find de, the side corresponding to ab

area_cde = area_abc - area_cde --> area_abc / area_cde = 2

let x=length of de

ratio of areas = (ratio of sides)^2

+2+=+%2836%2fx%29%5e2+

+2+=+1296%2fx%5e2+

++x%5e2+=+1296%2f2+=+648+

25.45584