the resultant force due to action of four forces is f1 which is 100N, along the negative y-axis.three of the force are 250N,320 above the x-axis,100N,60 above the x-axis, 200N, 140 above the x-axis find the fourth force
Correction: F1 = -100i
F5 = -100i-54.5i-88.3 = -88.3-154.5i
Tan Ar = -154.5/-88.3 = 1.74972
Ar = 60.3o = Reference angle.
A = 60.3 + 180 = 240.3o
F5 = X/cos A = -88.3/cos240.3 = 178.2 N.
[240.3o]
Resultant = F1 = 100N[270o].
F2 = 250N[320o].
F3 = 100N[60o].
F4 = 200N[140o].
F5 = ? Add to get F1.
X=250*cos320+100*cos60+200*cos140=88.3N
Y=250*sin320+100*sin60+200*sin140=54.5N
tan A = Y/X = 54.5/88.3 = 0.61680
A = 31.7o
F2+F3+F4=X/cos A = 88.3/cos 31.7 = 103.8N[31.7o] = 88.3 + 54.5i
F1 = 103.8[31.7o] + F5 = 100N[270o]
(88.3+54.5i) + F5 = 100i
F5 = 100i-54.5i-88.3 = -88.3 + 45.5i
Tan Ar = 45.5/-88.3 = -0.51529.
Ar = -27.3o
A = -27.3 + 180 = 152.7o
F5=X/cos A=-88.3/cos152.7=99.4N[152.7o]
To find the fourth force, we need to determine the sum of the forces in the y-direction as well as calculate the angle it makes with the negative y-axis.
Given forces:
F1 = 100 N (negative y-axis direction)
F2 = 250 N (above the x-axis)
F3 = 320 N (above the x-axis)
F4 = 100 N (above the x-axis)
F5 = 200 N (above the x-axis)
Let's calculate the resultant force in the y-direction:
Sum of forces above the x-axis = F2 + F3 + F4 + F5
= 250 N + 320 N + 100 N + 200 N
= 870 N
Since the resultant force is along the negative y-axis (downwards), we can say:
F1 = -870 N
Now, let's calculate the angle that the resultant force makes with the negative y-axis:
tan(theta) = F1 / Sum of forces above the x-axis
tan(theta) = -870 N / 870 N
tan(theta) = -1
From this, we can see that theta = -45 degrees, as tan(-45) = -1.
Therefore, the fourth force has a magnitude of 870 N and is directed at an angle of 45 degrees below the negative y-axis.
To find the fourth force, we need to determine the components of the three known forces along the y-axis and then find the sum of these components.
Let's break down each force into its y-component:
Force 1: F1 = 100 N along the negative y-axis.
Since it is acting directly along the negative y-axis, the y-component of F1 is simply 100 N.
Force 2: F2 = 250 N, 320 above the x-axis.
To find the y-component, we use the sine function:
y-component of F2 = F2 * sin(angle)
The angle is given as 320 above the x-axis, which means the complementary angle is 90 - 320 = -230 degrees.
Calculating the y-component of F2:
y-component of F2 = 250 N * sin(-230 degrees)
Force 3: F3 = 100 N, 60 above the x-axis.
Similar to Force 2, we calculate the y-component using the sine function:
y-component of F3 = F3 * sin(angle)
The angle is given as 60 above the x-axis, which means the complementary angle is 90 - 60 = 30 degrees.
Calculating the y-component of F3:
y-component of F3 = 100 N * sin(30 degrees)
Force 4: Unknown force (F4)
Since we know the resultant force, F1, is along the negative y-axis, the y-component of F4 must be negative to balance out F1. Therefore, we can write:
y-component of F4 = - (Sum of the y-components of F2, F3, and F4)
Finally, to find F4, we calculate the magnitude of the fourth force using its known y-component:
F4 = √(x-component^2 + y-component^2)
Substituting the calculated y-components into the formula, we can solve for F4.