Resultant Force Calculation: F1=800N, F2=1100N At 60°

Nov 4, 2025 by SLV Team 54 views

Hey guys! Let's dive into a physics problem that involves finding the resultant force when we have two forces acting at an angle. This is a common scenario in mechanics, and understanding how to solve it is crucial for anyone studying physics or engineering. We'll break down the steps to calculate the resultant force of F₁ = 800 N and F₂ = 1100 N acting at an angle of 60°. Get ready to put on your thinking caps!

Understanding Resultant Force

Before we jump into the calculations, let's make sure we're all on the same page about what resultant force actually means. Imagine you're pushing a box with two friends. Each of you is applying a force, but the box moves as if there's only one force acting on it. That single force that has the same effect as all the individual forces combined is the resultant force, also known as the net force. In simpler terms, it's the total force acting on an object.

When forces act in the same direction, you can simply add them up. If they act in opposite directions, you subtract them. But what happens when forces act at an angle, like in our problem? That's where things get a bit more interesting, and we need to use some trigonometry and the parallelogram law of vector addition. Don't worry, we'll walk through it step by step!

Knowing how to calculate resultant force is super important in many real-world situations. Think about designing bridges, buildings, or even analyzing the forces on a car during a crash. Understanding how forces combine helps engineers make structures safe and efficient. So, mastering this concept is definitely worth the effort!

Problem Setup: Forces at an Angle

Okay, let’s get our problem straight. We have two forces:

F₁ = 800 N
F₂ = 1100 N

These forces are acting at an angle (α) of 60° to each other. Our mission is to find the resultant force (let's call it Fᵣ) – that single force that represents the combined effect of F₁ and F₂. To do this, we'll use a method called the parallelogram law of vector addition. This might sound a bit intimidating, but it's actually a pretty neat trick.

The parallelogram law basically tells us that if we represent our forces as vectors (arrows with both magnitude and direction), we can construct a parallelogram with these vectors as adjacent sides. The diagonal of the parallelogram, starting from the point where the two vectors meet, represents the resultant force. The length of this diagonal gives us the magnitude of Fᵣ, and its direction gives us the direction of Fᵣ. Visually, it's like drawing a parallelogram with F₁ and F₂ as sides, and then the diagonal is our answer!

Before we dive into the formula, let’s quickly recap why we're doing this. We want to combine two forces acting at an angle into a single, equivalent force. This helps simplify complex systems and makes it easier to predict how an object will move or behave under the influence of multiple forces. Understanding the setup is half the battle, so we're off to a great start!

The Formula: Law of Cosines

Now for the fun part – the math! To calculate the magnitude of the resultant force (Fᵣ), we'll use a handy formula derived from the law of cosines. If you remember your trigonometry, the law of cosines relates the sides of a triangle to the cosine of one of its angles. In our case, the triangle is formed by the force vectors F₁, F₂, and the resultant force Fᵣ.

The formula looks like this:

Fᵣ = √ (F₁² + F₂² + 2 * F₁ * F₂ * cos(α))

Where:

Fᵣ is the magnitude of the resultant force.
F₁ and F₂ are the magnitudes of the individual forces (800 N and 1100 N in our case).
α is the angle between the forces (60°).
cos(α) is the cosine of the angle.

This formula might seem a bit daunting at first, but it's really just plugging in the numbers and doing the calculations. It’s like a magic recipe for finding the resultant force! The law of cosines is our key tool here, allowing us to handle the angled forces and get a precise result. Remember, the cosine of an angle is a value between -1 and 1, and it tells us how much the forces are aligned in the same direction.

Step-by-Step Calculation

Alright, let’s put that formula into action and calculate the resultant force for our problem. We'll break it down step-by-step to make it super clear.

Plug in the values:

Fᵣ = √ (800² + 1100² + 2 * 800 * 1100 * cos(60°))

Here, we've simply replaced F₁ with 800 N, F₂ with 1100 N, and α with 60°.
Calculate the squares:

Fᵣ = √ (640000 + 1210000 + 2 * 800 * 1100 * cos(60°))

We've squared 800 and 1100 to get 640000 and 1210000, respectively.
Find the cosine of 60°:

cos(60°) = 0.5

This is a common trigonometric value that you might want to memorize, but you can also use a calculator to find it.
Multiply the values:

Fᵣ = √ (640000 + 1210000 + 1760000 * 0.5)

We've multiplied 2 * 800 * 1100 to get 1760000.
Multiply by the cosine:

Fᵣ = √ (640000 + 1210000 + 880000)

Now, we multiply 1760000 by 0.5 to get 880000.
Add the values:

Fᵣ = √ (2730000)

Adding all the numbers inside the square root gives us 2730000.
Take the square root:

Fᵣ ≈ 1652.27 N

Finally, we take the square root of 2730000, which gives us approximately 1652.27 N.

So, the magnitude of the resultant force is approximately 1652.27 N. That’s a pretty significant force! We’ve successfully navigated the formula and arrived at our answer. High five!

Finding the Direction

Okay, we've figured out the magnitude (size) of the resultant force, but force is a vector, which means it also has a direction. We need to find the angle (let's call it θ) that the resultant force makes with one of the original forces, say F₁. To do this, we'll use another trigonometric relationship – the law of sines.

The law of sines states that in any triangle, the ratio of the length of a side to the sine of the angle opposite that side is constant. In our force parallelogram, this translates to:

sin(θ) / F₂ = sin(α) / Fᵣ

Where:

θ is the angle we want to find (the angle between F₁ and Fᵣ).
F₂ is the magnitude of the second force (1100 N).
α is the angle between F₁ and F₂ (60°).
Fᵣ is the magnitude of the resultant force (approximately 1652.27 N).

Let’s rearrange the formula to solve for sin(θ):

sin(θ) = (F₂ * sin(α)) / Fᵣ

Now, we plug in the values:

sin(θ) = (1100 * sin(60°)) / 1652.27

Calculating the Angle

Let's continue our quest to find the direction of the resultant force. We've already set up the equation using the law of sines, so now it's time to crunch some numbers.

Find the sine of 60°:

sin(60°) ≈ 0.866

This is another common trigonometric value. You can use a calculator or remember it from your trig knowledge.
Plug in the value:

sin(θ) = (1100 * 0.866) / 1652.27

We substitute sin(60°) with its approximate value.
Multiply and divide:

sin(θ) ≈ 952.6 / 1652.27

Multiplying 1100 by 0.866 gives us approximately 952.6.
Calculate the ratio:

sin(θ) ≈ 0.5765

Dividing 952.6 by 1652.27 gives us approximately 0.5765.
Find the angle θ:

θ = arcsin(0.5765)

To find the angle θ, we need to take the inverse sine (also known as arcsin or sin⁻¹) of 0.5765. This is the angle whose sine is 0.5765.
Use a calculator:

θ ≈ 35.23°

Using a calculator, we find that the arcsin of 0.5765 is approximately 35.23 degrees.

So, the resultant force acts at an angle of approximately 35.23° with respect to F₁. We've now determined both the magnitude and direction of the resultant force! That's a complete solution to our problem. Woohoo!

Summarizing the Result

Let's take a moment to recap what we've accomplished. We started with the problem of finding the resultant force of two forces, F₁ = 800 N and F₂ = 1100 N, acting at an angle of 60°. We broke down the problem into manageable steps, using the parallelogram law of vector addition and the law of cosines to find the magnitude of the resultant force, and the law of sines to find its direction.

Here's what we found:

Magnitude of the Resultant Force (Fᵣ): Approximately 1652.27 N
Direction of the Resultant Force (θ): Approximately 35.23° with respect to F₁

This means that the combined effect of the two forces is equivalent to a single force of about 1652.27 N acting at an angle of about 35.23° relative to the 800 N force. That's pretty cool, right?

Understanding how to calculate resultant forces is a fundamental skill in physics and engineering. It allows us to analyze and predict the behavior of objects under the influence of multiple forces. Whether you're designing a bridge, analyzing the motion of a projectile, or simply trying to understand how forces combine in everyday situations, this knowledge will come in handy.

Practical Applications and Further Learning

Now that we've mastered this calculation, let's think about where this knowledge can be applied in the real world. Understanding resultant forces is crucial in various fields:

Engineering: When designing structures like bridges and buildings, engineers need to calculate the forces acting on different components to ensure stability and safety. They use the principles we've discussed to determine the resultant force and ensure that the structure can withstand the loads.
Physics: Studying the motion of objects often involves analyzing the forces acting on them. Calculating the resultant force helps predict the acceleration and trajectory of objects, whether it's a ball thrown in the air or a satellite orbiting the Earth.
Sports: In sports like archery or golf, understanding resultant forces can help athletes optimize their technique. The force they apply and the angle at which they apply it directly affect the trajectory of the projectile (arrow or ball).
Everyday Life: Even in everyday situations, we intuitively deal with resultant forces. When you push a heavy piece of furniture with someone else, the combined force you apply is the resultant force.

If you're interested in diving deeper into this topic, there are many resources available. You can explore textbooks on mechanics, online courses, and interactive simulations. Practice solving different problems with varying forces and angles to solidify your understanding. The more you practice, the more comfortable you'll become with these concepts. Keep exploring, keep learning, and you'll become a force to be reckoned with in physics!

So, there you have it, guys! We've successfully calculated the resultant force of two forces acting at an angle. Remember, practice makes perfect, so keep those calculations coming! You've got this!