An object in equilibrium has three forces exerted on it. A 36 N force acts at 90° from the x axis and a 42 N force acts at 60°. What are the magnitude and direction of the third force?
__?__N
__?__° (counterclockwise from the +x direction)
To find the magnitude and direction of the third force, we need to use the concept of vector addition and equilibrium conditions.
First, let's draw a diagram to visualize the forces:
F1 (36 N)
/
/
/
/
/
/
/
_____________/
\
\
\ F2 (42 N)
The x-axis is horizontal, and the y-axis is vertical.
Now, let's break down each force into its x and y components:
F1 = 36 N at 90°:
Fx1 = 0 N (no x-component)
Fy1 = 36 N
F2 = 42 N at 60°:
Fx2 = 42 N * cos(60°) = 21 N
Fy2 = 42 N * sin(60°) = 36.372 N
Since the object is in equilibrium, the net force in both the x and y directions must be zero. Therefore, the x and y components of the third force must balance out the x and y components of the given forces.
Net Fx = Fx1 + Fx2 + Fx3 = 0
Net Fy = Fy1 + Fy2 + Fy3 = 0
Since Fx1 and Fy1 are both zero, the third force must have an x-component of -21 N and a y-component of -36.372 N to cancel out Fx2 and Fy2, respectively.
Fx3 = -21 N
Fy3 = -36.372 N
To find the magnitude of the third force, we can use the Pythagorean theorem since the magnitude of the force is the hypotenuse of the triangle formed by Fx3 and Fy3:
|F3| = sqrt(Fx3^2 + Fy3^2)
|F3| = sqrt((-21 N)^2 + (-36.372 N)^2)
|F3| ≈ 42.55 N
Therefore, the magnitude of the third force is approximately 42.55 N.
To find the direction of the third force, we can use the inverse tangent function (tan^(-1)):
θ = tan^(-1)(Fy3 / Fx3)
θ = tan^(-1)(-36.372 N / -21 N)
θ ≈ 60.84°
Since the direction is counterclockwise from the +x direction, we can subtract this angle from 180° to obtain the direction:
θ = 180° - 60.84°
θ ≈ 119.16°
Therefore, the direction of the third force is approximately 119.16° counterclockwise from the +x direction.
The third force has a magnitude of approximately 42.55 N and a direction of approximately 119.16° counterclockwise from the +x direction.