# Finding magnitude and direction of equilibrant.

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In summary, the magnitude and direction of the equilibrant of two forces can be found using different methods such as the Pythagorean theorem, Soh Cah Toa, and the Cosine law. The correct direction can be found by subtracting the calculated angle from 180 degrees.
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## Homework Statement

Find the magnitude and direction of the equilibrant of each of the following foces:

a) forces of 32N and 48N acting at an angle of 90 degress to each other
b) forces of 16N and 10N acting at an angle of 19 degrees to each other

## Homework Equations

Sine Law, Cosine law, Soh Cah Toa, Pythagorean.

## The Attempt at a Solution

a) Because there is a 90 degree angle, to find the missing side created to make a triangle with the two forces, I would use the pythagorean theorum. The square root of 32^2+48^2 is 57.7 Newtons. This I know is correct. However, when trying to find the direction, I've become puzzled. Since it is a right angled triangle, I used soh cah toa, in which: tanx=(32/48), x=34 degrees. So the direction would be 34 degrees to 32N, correct? The answer page is 146 to 48 N. I'm not sure if there can be two correct answers, as I have found that 180 subtract my original answer of 34 would give me 146, the "correct" answer. Could someone explain this to me?

b) Since there is no right angle, I am using the Cosine law. a^2=10^2 + 16^2-2(10)(16)cos170 (By creating a parallelogram, I found the angle opposite to a is 180-10 degrees). The answer I get is 25.9N. Again, this is correct, and again, I've no clue why my direction is not the same as the answer page.

Using the sine law, sinx/16N = sin170/26. The answer is 6.1 degrees to 16N. The back of the book says 174 degrees to 10N. Again, I have found that 180 subtract my answer of 6.1 degrees would give me 174, the correct answer. I have absolutely no idea why I must subtract by 180 to find the answer. Please, anyone care to explain to me?

This is part of the Calculus and Vectors course, but since this is vectors, I figured it would be more appropriate to post it in the Physics section,thanks.

EDIT: Actually, my teacher advised the physics method is different, so if somone can move this to the Calculus section, you are appreciated.

There are several ways to add them.

One is to resolve them into components i,j and add the components and resolve the Resulting vector.

The other head to tail addition, which makes the parallelogram like you were doing. It all should come out to the same result.
I am using the Cosine law. a^2=10^2 + 16^2-2(10)(16)cos170
On this one I don't understand your angle.
If they are 19° from each other I would think the angle was 161° if you were adding 1 tail to head of the other.

I would like to clarify the concept of equilibrant and its calculation in this context. The equilibrant is a single force that can balance out the combined effect of multiple forces acting on an object. In order to find the magnitude and direction of the equilibrant, we need to first understand the concept of vector addition.

In vector addition, the resultant vector is the sum of all the individual vectors. This can be represented by a triangle, where the individual vectors are the sides of the triangle and the resultant vector is the diagonal. The magnitude and direction of the resultant vector can be calculated using the Pythagorean theorem and trigonometric functions respectively.

In the case of a), the two forces of 32N and 48N are acting at right angles to each other. This means that the resultant vector is the hypotenuse of a right triangle, and its magnitude can be calculated using the Pythagorean theorem. However, the direction of the resultant vector is not 34 degrees as calculated using the tangent function. This is because the tangent function gives the angle between the resultant vector and the adjacent side of the triangle, which in this case is the 32N force. The correct way to find the direction is to use the inverse tangent function, which in this case gives us an angle of 146 degrees with respect to the 48N force.

In the case of b), the two forces of 16N and 10N are not acting at right angles to each other. Therefore, we cannot use the Pythagorean theorem to find the magnitude of the resultant vector. Instead, we need to use the cosine law, as you have correctly done. However, the angle you have calculated (170 degrees) is not the angle between the resultant vector and the 10N force. Again, the correct way to find the angle is to use the inverse cosine function, which gives us an angle of 174 degrees with respect to the 10N force.

In summary, the key to finding the direction of the equilibrant is to use the inverse trigonometric functions and to be careful in identifying the angle between the resultant vector and the individual force. I hope this explanation helps clarify the concept of equilibrant and its calculation.

## What is the definition of equilibrant?

The equilibrant is a force that is equal in magnitude and opposite in direction to the resultant force acting on an object. It is the force that would be needed to counteract the resultant force and keep the object in a state of equilibrium.

## How do you find the magnitude of an equilibrant?

To find the magnitude of an equilibrant, you can use the Pythagorean theorem. First, find the horizontal and vertical components of the resultant force. Then, use these components to calculate the magnitude of the equilibrant using the formula c = √(a² + b²), where c is the magnitude of the equilibrant and a and b are the horizontal and vertical components, respectively.

## What is the significance of finding the equilibrant?

Finding the equilibrant is important because it tells us the minimum amount of force needed to keep an object in equilibrium. It also helps us understand the relationship between the different forces acting on an object and how they contribute to the object's overall motion.

## How do you determine the direction of an equilibrant?

The direction of the equilibrant is always opposite to the direction of the resultant force. This can be determined by drawing a free body diagram of the object and labeling the forces acting on it. The equilibrant will be drawn in a direction that is opposite to the direction of the resultant force.

## What are some real-world applications of finding the equilibrant?

The concept of equilibrant is used in many fields, including engineering, physics, and architecture. It is essential for understanding the stability and balance of structures such as bridges, buildings, and machines. It is also used in designing systems that require precise balancing, such as gyroscopes and aircraft.

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