A bolt secured to a bridge is being acted upon by forces being exerted by two cables attached to the bolt. The force vector for one of the cables is represented by F1 = <1.7,9.8>. The other force vector, F2, is known to have a magnitude 5 and a direction angle of 10 degrees. Determine the angle between the two cables that are attached to this bolt. (Round your answer to 2 decimal places.)
convert the first from rectangular to polar coordinates:
r^2 = 1.7^2 + 9.8^2 = 98.93
r = 9.95
tanθ = 9.8/1.7
θ = 80.16°
so, the angle between the vectors is 80.16 - 10 = 70.16°
To determine the angle between the two cables, we need to use the concept of vector addition. In vector addition, the resultant vector is found by adding the individual components of the vectors.
First, let's find the components of the second force vector, F2. We are given the magnitude (5) and the direction angle (10 degrees). We can find the x-component and y-component of F2 using trigonometry.
The x-component, F2x, is given by:
F2x = magnitude * cos(angle)
F2x = 5 * cos(10)
The y-component, F2y, is given by:
F2y = magnitude * sin(angle)
F2y = 5 * sin(10)
Now, we have both force vectors represented in terms of their components:
F1 = <1.7, 9.8>
F2 = <F2x, F2y>
To find the angle between the two cables, we can use the dot product formula:
cos(theta) = (F1 dot F2) / (|F1| * |F2|)
The dot product of two vectors is found by multiplying their corresponding components and then adding up the results:
F1 dot F2 = 1.7 * F2x + 9.8 * F2y
The magnitude of a vector is found using the Pythagorean theorem:
|F1| = sqrt(1.7^2 + 9.8^2)
|F2| = sqrt(F2x^2 + F2y^2)
Finally, we can substitute all these values into the equation for cos(theta) to find the angle between the two cables:
cos(theta) = (1.7 * F2x + 9.8 * F2y) / (sqrt(1.7^2 + 9.8^2) * sqrt(F2x^2 + F2y^2))
Let's calculate the values and find the angle.