Two forces 10 N acting in the direction N 30 east and 15 N acting in the eastern direction, if both forces act at a point, find the magnitude and direction of (a) the resultant force (b) the equilibriant force.

a. Fr = 10N[60o]CCW + 15N[0o]

X = 10*Cos60 + 15 = 20 N.
Y = 10*sin60 = 8.7 N.
Q1.

Tan A = Y/X = 8.7/20 = 0.433
A = 23.4o N. of E.

Fr = X/Cos A = 20/Cos23.4=21.8 N.[23.4o]
N. of E. = 23.4o CCW

b. Equilibrant:
X = -20
Y = -8.7
Q3.

Tan A = Y/X = -8.7/-20 = 0.433
A = 23.4o S. of W. = 203.4o CCW.

Fr = X/Cos A = -20/Cos203.4 = 21.8 N[23.4o S. of W. = 203.4o CCW.

Note: 23.4o S. of W. is the opposite of
23.4o N. of E. and their magnitudes are
equal.

Diagram

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I need a diagram

I want a clear solution because you didn't involved N30 from the question above

To find the magnitude and direction of the resultant force, you can use the concept of vector addition.

Step 1: Resolve each force into its horizontal and vertical components.

Given:
Force 1: 10 N acting at a direction of 30° east
Force 2: 15 N acting in the eastern direction

For Force 1:
Horizontal Component: F₁ₓ = 10 N * cos(30°) = 10 N * √3/2 = 5√3 N
Vertical Component: F₁ᵧ = 10 N * sin(30°) = 10 N * 1/2 = 5 N

For Force 2:
Since the force is acting directly east, there is no vertical component. The entire force is horizontal.
Horizontal Component: F₂ₓ = 15 N

Step 2: Add the horizontal and vertical components separately to find the resultant horizontal and vertical components.

Resultant Horizontal Component: Rₓ = F₁ₓ + F₂ₓ = 5√3 N + 15 N = 5√3 N + 15 N = 5√3 N + 15 N = 5√3 + 15 N = 15 + 5√3 N

Resultant Vertical Component: Rᵧ = F₁ᵧ + 0 = 5 N + 0 = 5 N

Step 3: Use the Pythagorean theorem to find the magnitude of the resultant force.

Magnitude of Resultant Force (R): R = √(Rₓ² + Rᵧ²)
R = √((15 + 5√3)² + 5²)
R = √(225 + 150√3 + 75 + 25)
R = √(325 + 150√3)

Step 4: Use trigonometry to find the direction of the resultant force.

Direction of Resultant Force: θ = tan⁻¹(Rᵧ/Rₓ)
θ = tan⁻¹(5/ (15 + 5√3))

To find the magnitude and direction of the equilibrant force, you need to consider that the equilibrant force has the same magnitude as the resultant force but acts in the opposite direction.

Magnitude of Equilibrant Force: E = R

Direction of Equilibrant Force: Opposite direction of the resultant force (180° + θ)

So, using the above steps, you can calculate the magnitude and direction of the resultant force and the equilibrant force.