Two forces are exerted on an object. A 33 N force acts at 215° and a 45 N force acts at 315°. What are the magnitude and direction of the equilibrant?
F1 + F2 + F3 = 0
33N[215o] + 45N[315] + F3 = 0
33*cos215+i33*sin215+45*cos315+i45*sin315+F3 = 0
23.33 - i18.93 + 31.82 - i31.82 + F3=0
55.15 - i50.75 + F3 = 0
F3=-55.15 + i50.75=sqr(-55.15^2+50.75^2)= 74.95 N. = The equilibrant.
Tan Ar = Y/X = 50.75/-55.15=-0.92022
Ar = -42.62o = Reference angle.
A = -42.62 + 180 = 137.4o = Direction.
To find the magnitude and direction of the equilibrant, we need to first determine the resultant force of the two given forces. The equilibrant is a force that, when added to the original forces, will create a net force of zero (thus bringing the object into equilibrium).
To find the resultant force, we can use vector addition. We add the given forces together by breaking them down into their horizontal and vertical components.
First, let's find the horizontal components of the forces:
Force 1 (33 N at 215°):
Horizontal component = Force 1 * cos(215°) = 33 N * cos(215°) ≈ -27.19 N
Force 2 (45 N at 315°):
Horizontal component = Force 2 * cos(315°) = 45 N * cos(315°) ≈ 31.82 N
Next, let's find the vertical components of the forces:
Force 1 (33 N at 215°):
Vertical component = Force 1 * sin(215°) = 33 N * sin(215°) ≈ -11.52 N
Force 2 (45 N at 315°):
Vertical component = Force 2 * sin(315°) = 45 N * sin(315°) ≈ -31.82 N
Now, we can add the horizontal and vertical components of the forces:
Resultant horizontal component = Horizontal component of Force 1 + Horizontal component of Force 2 ≈ -27.19 N + 31.82 N ≈ 4.63 N
Resultant vertical component = Vertical component of Force 1 + Vertical component of Force 2 ≈ -11.52 N + (-31.82 N) ≈ -43.34 N
To find the magnitude of the resultant force (R), we can use the Pythagorean theorem:
R = √(Resultant horizontal component^2 + Resultant vertical component^2)
= √(4.63 N^2 + (-43.34 N)^2)
≈ √(21.43 N^2 + 1875.31 N^2)
≈ √1896.74 N^2
≈ 43.56 N
Therefore, the magnitude of the equilibrant is approximately 43.56 N.
To find the direction of the equilibrant, we can use trigonometry. We can find the angle using the arctangent function:
Angle of the equilibrant = arctan(Resultant vertical component / Resultant horizontal component)
= arctan(-43.34 N / 4.63 N)
≈ arctan(-9.36)
Since the angle is negative, we need to add 180° to find the reference angle:
Reference angle = Angle of the equilibrant + 180°
≈ arctan(-9.36) + 180°
Using a calculator, we find that the reference angle is approximately 172.15°.
Since the resultant vector is in the fourth quadrant (positive horizontal, negative vertical), the direction of the equilibrant is 180° - reference angle:
Direction of the equilibrant ≈ 180° - 172.15°
≈ 7.85° (rounded to two decimal places)
Therefore, the magnitude of the equilibrant is approximately 43.56 N, and its direction is approximately 7.85°.