Finding magnitude and direction of equilibrant.

[Obama]

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.

 

[LowlyPion]

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.

 

[nlightner]

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.