Forces of 5N in three places act at an angle of 30°.Find the resultant force and the angle of resultant that makes with the horizontal.
Forces of 5N in three places act at an angle of 30°
what does that even mean?
To find the resultant force and the angle it makes with the horizontal, we need to use vector addition.
Step 1: Break down the given forces into their horizontal and vertical components.
Since the angle between the forces is given as 30°, we can determine the horizontal and vertical components using trigonometry.
Horizontal Component = Force * cos(Angle)
Vertical Component = Force * sin(Angle)
Let's calculate the horizontal and vertical components of each force:
For Force 1:
Horizontal Component = 5N * cos(30°)
Vertical Component = 5N * sin(30°)
For Force 2:
Horizontal Component = 5N * cos(30°)
Vertical Component = 5N * sin(30°)
For Force 3:
Horizontal Component = 5N * cos(30°)
Vertical Component = 5N * sin(30°)
Step 2: Add up the horizontal and vertical components separately.
To find the resultant force, we need to add up the horizontal components and the vertical components.
Horizontal Resultant = Sum of Horizontal Components
Vertical Resultant = Sum of Vertical Components
Horizontal Resultant = (Horizontal Component of Force 1) + (Horizontal Component of Force 2) + (Horizontal Component of Force 3)
Vertical Resultant = (Vertical Component of Force 1) + (Vertical Component of Force 2) + (Vertical Component of Force 3)
Step 3: Calculate the magnitude of the resultant force using Pythagoras theorem.
The magnitude of the resultant force can be found using the formula:
Resultant Magnitude = sqrt((Horizontal Resultant)^2 + (Vertical Resultant)^2)
Step 4: Calculate the angle of the resultant force.
The angle the resultant force makes with the horizontal can be determined using trigonometry:
Resultant Angle = arctan(Vertical Resultant / Horizontal Resultant)
Let's plug in the values and calculate the answer:
Horizontal Component = 5N * cos(30°) = 5N * √3/2 ≈ 4.33N
Vertical Component = 5N * sin(30°) = 5N * 1/2 = 2.5N
Horizontal Resultant = 4.33N + 4.33N + 4.33N = 13N
Vertical Resultant = 2.5N + 2.5N + 2.5N = 7.5N
Resultant Magnitude = sqrt((13N)^2 + (7.5N)^2) ≈ sqrt(169N^2 + 56.25N^2) ≈ sqrt(225.25N^2) ≈ 15N
Resultant Angle = arctan(7.5N / 13N) ≈ arctan(0.577) ≈ 29.7°
Therefore, the resultant force has a magnitude of 15N and makes an angle of approximately 29.7° with the horizontal.