An object is acted upon by two forces operating 5N and 12 N. calculate the resultant of the force acting on an angle of 120 degree to each other.
if the 5N force acts in the +i direction, then the resultant is
r = 5i + (-6 i + 6√3 j) = -i + 6√3j
|r| = √(1^2 + 108) = √109
at an angle θ where
tanθ = 6√3/-1 = -6√3
θ = 95.5°
Well, when you have two forces acting on an object, you can find the resultant force by using the concept of vector addition. However, since you mentioned the forces are at an angle of 120 degrees to each other, I'll have to bring out my trusty clown calculator for this one!
Now, let's imagine the 5N force as a cheerful little clown named Forcey, and the 12N force as a jolly jester called Jester Jolty. These two forces will ultimately create a resultant force, who we'll name Sir Resultant.
To find Sir Resultant, we'll use some mathematical magic! We'll need to break down the forces into their x and y components. The 5N force will have a component in the x-direction, while the 12N force will have components in both the x and y directions.
Using some trigonometric tricks, we can calculate that the x-component of the 5N force is 5N * cos(120°), which is -2.5N (since cos(120°) = -0.5). The y-component of the 12N force is 12N * sin(120°), which is 10.39N (since sin(120°) = 0.866).
Now, we can add up the x-components and y-components separately. The x-components of the forces are -2.5N (from Forcey) and 12N * cos(120°) = -6N (from Jester Jolty). When we add them up, we get -8.5N.
For the y-components, we have 10.39N (from Jester Jolty), and the y-component of the 5N force is 5N * sin(120°) = 4.33N (since sin(120°) = 0.866 again). When we add them up, we get 14.72N.
Now it's time for the grand finale! Using the x and y components, we can calculate the magnitude of the resultant force using the Pythagorean theorem. The magnitude of the resultant force is √((-8.5N)² + (14.72N)²), which comes out to be approximately 17.1N.
So, the magnitude of the resultant force is around 17.1N. But remember, the direction of the resultant force is equally important! Why don't you give me a compass, and we'll figure that out too?
Just kidding! The angle of the resultant force can be found using trigonometry once again. It's ∠θ = tan^(-1)((14.72N)/(-8.5N)), which comes out to be approximately 118.7°.
So, to sum it up, the resultant force has a magnitude of around 17.1N and acts at an angle of approximately 118.7°. Sir Resultant has entered the chat, ready to clown around!
To calculate the resultant of two forces acting on an angle, we can use the following steps:
Step 1: Break down the forces into their components.
Step 2: Find the x-components and y-components of the forces.
Step 3: Calculate the sum of the x-components and the sum of the y-components.
Step 4: Use the sum of the x-components (ΣFx) and the sum of the y-components (ΣFy) to find the resultant force (R) using the Pythagorean theorem: R = √(ΣFx² + ΣFy²)
Step 5: Determine the angle (θ) of the resultant force using the inverse tangent function: θ = tan⁻¹(ΣFy / ΣFx)
Now let's apply these steps to calculate the resultant force for the given scenario:
Step 1: The two forces operating on the object are 5N and 12N, acting at an angle of 120 degrees to each other.
Step 2: Break down the forces into their x (horizontal) and y (vertical) components:
Force 1 (5N):
Fx1 = 5N × cos(120°) = -2.5N (Negative because it points to the left)
Fy1 = 5N × sin(120°) = 4.33N (Positive because it points upwards)
Force 2 (12N):
Fx2 = 12N × cos(120°) = -6N (Negative because it points to the left)
Fy2 = 12N × sin(120°) = 10.39N (Positive because it points upwards)
Step 3: Calculate the sum of the x-components and the sum of the y-components:
ΣFx = Fx1 + Fx2 = -2.5N + (-6N) = -8.5N
ΣFy = Fy1 + Fy2 = 4.33N + 10.39N = 14.72N
Step 4: Use the Pythagorean theorem to find the resultant force:
R = √(ΣFx² + ΣFy²) = √((-8.5N)² + (14.72N)²) = √(72.25N² + 216.81N²)
R = √289.06N²
R = 17N (approximately)
Step 5: Determine the angle of the resultant force:
θ = tan⁻¹(ΣFy / ΣFx) = tan⁻¹(14.72N / -8.5N) ≈ -59.47°
Therefore, the resultant force acting on the object is approximately 17N and is at an angle of -59.47° to the x-axis.
To calculate the resultant of two forces acting at an angle, we can use vector addition.
First, we need to break down each force into its components along the x and y axes. In this case, we can assume the x-axis is the horizontal axis and the y-axis is the vertical axis.
Let's calculate the x and y components for each force:
Force 1 (5N) at an angle of 120 degrees:
F1x = 5N * cos(120°)
= -5N / 2 (taking the negative sign because it's in the opposite direction of the positive x-axis)
F1y = 5N * sin(120°)
= 5N * √3 / 2
Force 2 (12N) at an angle of 120 degrees:
F2x = 12N * cos(120°)
= -12N / 2
= -6N
F2y = 12N * sin(120°)
= 12N * √3 / 2
Now, calculate the resultant force by adding the x and y components separately:
R = Rx + Ry
Rx = F1x + F2x
= (-5N / 2) + (-6N)
= -5N/2 - 6N/1
= -16N/2
= -8N
Ry = F1y + F2y
= (5N * √3 / 2) + (12N * √3 / 2)
= (5√3N + 12√3N) / 2
= (17√3N) / 2
Now, calculate the magnitude of the resultant force using the Pythagorean theorem:
|R| = sqrt(Rx^2 + Ry^2)
= sqrt((-8N)^2 + ((17√3N) / 2)^2)
Simplifying the expression:
|R| = sqrt(64N^2 + (289/4) * 3N^2)
= sqrt(64N^2 + (867/4)N^2)
= sqrt((256 + 867) / 4) * N
= sqrt(1123 / 4) * N
= sqrt(280.75) * N
≈ 16.76N
Therefore, the magnitude of the resultant force is approximately 16.76N.