A force of 369 lbs makes an angle of 18 degrees 20 minutes with a force 427 lbs. Find the angle made by the equilibrant with the 369 lbs force.
I know I can find this using the law of cosines, but I think my teacher wants me to use some vector thing, but I do not know how to relate the two vectors to the equilibrant. Is it like u-v?
Hmmm. use the 427 Lbs at 0 degrees, then the cos18'20" * 369 is the force in the direction of zero degrees, and 369*sin18'20"is the force 90 degrees to that. Now add these two to see the resulatant, and then add180 degrees to get the equilibrant.
Well, either way it is a "vector thing.
The "equilibrant" is equal and opposite to the "resultant" or sum of the two vectors. In other words it is the vector needed to cancel them.
I will find the resultant, then we can do minus signs.
I am going to say we have 369 pounds along the x axis and 427 pounds at 18 1/3 degrees above the x axis.
In vector notation that is
Rx = 369 + 427 cos 18.33
Ry = 427 sin 18.33
or
Rx = 369 + 405 = 774
Ry = 134
Our equilibrant has components then
Ex = -774
Ey = -134
That is in the third quadrant at
tan A = 134/774 where A is below the -x axis
A = 9.82 degrees below -x axis
or
189.8 degrees counterclockwise from x axis where our original 369 lb force was
Yes, you can find the angle made by the equilibrant with the 369 lbs force using vector subtraction. The equilibrant is the vector that, when added to the given force, results in a net force of zero.
To relate the two vectors involved (369 lbs and 427 lbs) to the equilibrant, you can use the concept of vector addition and subtraction. In this case, you need to subtract the given force (369 lbs) from the other force (427 lbs) to find the equilibrant.
Let's denote the given force as vector A (369 lbs) and the other force as vector B (427 lbs). The equilibrant, denoted by vector E, can be found by subtracting vector A from vector B:
E = B - A
To find the angle made by the equilibrant with the 369 lbs force, you can use the dot product. The dot product of two vectors A and B is calculated as:
A · B = |A| |B| cos(theta)
Where |A| and |B| are the magnitudes (lengths) of vectors A and B respectively, and theta is the angle between them.
In this case, you need to find the angle between vector E and vector A. The dot product of vector E and vector A can be written as:
E · A = |E| |A| cos(theta_equilibrant_A)
Since the net force is zero when the given force and the equilibrant are added together, the dot product of E and A is zero:
E · A = 0
So, you can set up the equation:
0 = |E| |A| cos(theta_equilibrant_A)
Since |A| is known as 369 lbs, and |E| is the magnitude of the equilibrant you are looking for, you can solve for theta_equilibrant_A.
cos(theta_equilibrant_A) = 0 / (|E| |A|)
= 0 / (|E| * 369 lbs)
Since cos(theta) = 0 when theta is 90 degrees, the equation becomes:
(|E| * 369 lbs) = 0
To solve for |E|, the magnitude of the equilibrant, you need more information (such as the value of theta_equilibrant_A or the magnitude of the other force B). Without that information, it is not possible to determine the angle made by the equilibrant with the 369 lbs force using vector subtraction alone.
To find the angle made by the equilibrant with the 369 lbs force, we can indeed use vector addition. The equilibrant is the vector that, when added to the given forces, results in the vector sum of zero.
Let's represent the forces as vectors. We can call the force of 369 lbs as vector A, the force of 427 lbs as vector B, and the equilibrant force as vector C.
Now, to find vector C, we need to add vector A and vector B. Since vector C will result in a zero vector sum, its magnitude should be the same as the vector sum of A and B, but with the opposite direction.
To add the vectors, we need to break them down into their components. We can use the following formulas:
Ax = A * cos(θ)
Ay = A * sin(θ)
Bx = B * cos(ϕ)
By = B * sin(ϕ)
Here, θ is the angle between vector A and the x-axis, and ϕ is the angle between vector B and the x-axis. By finding the values of Ax, Ay, Bx, and By, we can now add the components to find the components of vector C:
Cx = Ax + Bx
Cy = Ay + By
To find the magnitude and angle of vector C, we can use these components:
C = sqrt(Cx^2 + Cy^2)
ψ = arctan(Cy / Cx)
Finally, the angle made by the equilibrant with the 369 lbs force is the complement of ψ. So, the angle you're looking for is 90 degrees minus ψ.
Keep in mind that the angles should be in the same units (either degrees or radians) throughout the calculations.