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.
F1=10 N at 60°
F2=15 N at 0°
Use x- and y-components method:
x-component
Fx=F1*cos(60)+F2*cos(0)
=20 N
y-component
Fy=F1*sin(60)+F2*sin(0)
=8.660254 N
Resultant, R= √(Fx²+Fy²)
angle, θ = atan2(Fy, Fx)
Hints for checking your answer:
20 N < R <25 N
20°< θ <25°
To find the magnitude and direction of the resultant force and the equilibrant force, we can use vector addition and subtraction.
(a) Resultant Force:
Step 1: Resolve the forces into their components.
The force of 10 N acting at N 30° east can be resolved into horizontal and vertical components as follows:
Horizontal Component = 10 N * cos(30°) = 10 N * 0.866 = 8.66 N
Vertical Component = 10 N * sin(30°) = 10 N * 0.5 = 5 N
The force of 15 N acting in the eastern direction can be considered as already in the horizontal direction.
Step 2: Add the horizontal and vertical components individually.
Horizontal Component: 8.66 N + 15 N = 23.66 N
Vertical Component: 5 N
Step 3: Use the Pythagorean theorem to find the magnitude of the resultant force.
Resultant Force = sqrt((Horizontal Component)^2 + (Vertical Component)^2)
Resultant Force = sqrt((23.66 N)^2 + (5 N)^2)
Resultant Force = sqrt(558.4356 N^2 + 25 N^2)
Resultant Force = sqrt(583.4356 N^2)
Resultant Force ≈ 24.15 N
Step 4: Find the direction of the resultant force.
Tan(θ) = Vertical Component / Horizontal Component
θ = tan^(-1)(Vertical Component / Horizontal Component)
θ = tan^(-1)(5 N / 23.66 N)
θ ≈ 12.05°
Therefore, the magnitude of the resultant force is approximately 24.15 N, and the direction is approximately N 12.05° east.
(b) Equilibrant Force:
The equilibrant force is the force required to balance the resultant force. It points in the opposite direction of the resultant force.
Magnitude of the Equilibrant Force = Magnitude of the Resultant Force = 24.15 N
Direction of the Equilibrant Force = Opposite direction of the Resultant Force = N 12.05° west.
To find the magnitude and direction of the resultant force and the equilibrant force, we can use vector addition and subtraction.
(a) Resultant Force:
To find the magnitude and direction of the resultant force, we need to add the two given forces.
Step 1: Convert the given forces into their x and y components.
- The force of 10 N acting in the direction N 30° East can be split into x and y components as follows:
- x-component = 10 N * cos(30°) = 8.66 N
- y-component = 10 N * sin(30°) = 5 N
- The force of 15 N acting in the eastern direction (which is already along the x-axis) has an x-component of 15 N and a y-component of 0 N.
Step 2: Add the x-components and the y-components separately to find the resultant components.
- Resultant x-component = 8.66 N + 15 N = 23.66 N
- Resultant y-component = 5 N + 0 N = 5 N
Step 3: Calculate the magnitude of the resultant force using the Pythagorean theorem.
- Magnitude of the resultant force = sqrt((Resultant x-component)^2 + (Resultant y-component)^2)
= sqrt((23.66 N)^2 + (5 N)^2)
≈ 24 N (rounded to two decimal places)
Step 4: Calculate the direction of the resultant force.
- Direction of the resultant force = arctan(Resultant y-component / Resultant x-component)
= arctan(5 N / 23.66 N)
≈ 12.04° (rounded to two decimal places)
Therefore, the direction of the resultant force is approximately 12.04° north of east.
So, the magnitude of the resultant force is approximately 24 N, and its direction is approximately 12.04° north of east.
(b) Equilibrant Force:
To find the magnitude and direction of the equilibrant force, we need to subtract the given forces.
Step 1: Convert the given forces into their x and y components (already done above).
Step 2: Subtract the x-components and the y-components separately to find the equilibrant components.
- Equilibrant x-component = 15 N - 8.66 N = 6.34 N
- Equilibrant y-component = 0 N - 5 N = -5 N
Step 3: Calculate the magnitude of the equilibrant force using the Pythagorean theorem.
- Magnitude of the equilibrant force = sqrt((Equilibrant x-component)^2 + (Equilibrant y-component)^2)
= sqrt((6.34 N)^2 + (-5 N)^2)
≈ 8.06 N (rounded to two decimal places)
Step 4: Calculate the direction of the equilibrant force.
- Direction of the equilibrant force = arctan(Equilibrant y-component / Equilibrant x-component)
= arctan(-5 N / 6.34 N)
≈ -37.03° (rounded to two decimal places)
Therefore, the direction of the equilibrant force is approximately 37.03° south of west.
So, the magnitude of the equilibrant force is approximately 8.06 N, and its direction is approximately 37.03° south of west.