Cable AC is 70 ft long, and tension in the cable is 5000 lb. Determine (a) the x, y, and z components of the force exerted by the cable on the anchor C.
(b) the angles theta x, theta y, and theta z defining the direction of that force.
To determine the x, y, and z components of the force exerted by the cable on the anchor C, we can use the following formulas:
(a) x-component of the force (Fₓ):
Fₓ = T * cos(thetaₓ)
(b) y-component of the force (Fᵧ):
Fᵧ = T * cos(thetaᵧ)
(c) z-component of the force (F_z):
F_z = T * cos(theta_z)
where T is the tension in the cable (5000 lb).
To find the angles thetaₓ, thetaᵧ, and theta_z, we can use the following formulas:
(d) tan(thetaₓ) = y/z
==> thetaₓ = arctan(y/z)
(e) tan(thetaᵧ) = x/z
==> thetaᵧ = arctan(x/z)
(f) tan(theta_z) = sqrt(x² + y²) / z
==> theta_z = arctan(sqrt(x² + y²) / z)
Now, let's calculate the components of the force and the angles:
Step 1: Find the lengths of x, y, and z components using the Pythagorean theorem.
Given:
- Cable length (AC) = 70 ft
- Tension in the cable (T) = 5000 lb
Using the Pythagorean theorem:
z² = AC² = 70²
z = sqrt(70²)
z = 70 ft
Step 2: Calculate the components of the force:
(a) x-component (Fₓ):
Fₓ = T * cos(thetaₓ)
Fₓ = 5000 * cos(thetaₓ)
(b) y-component (Fᵧ):
Fᵧ = T * cos(thetaᵧ)
Fᵧ = 5000 * cos(thetaᵧ)
(c) z-component (F_z):
F_z = T * cos(theta_z)
F_z = 5000 * cos(theta_z)
Step 3: Calculate the angles thetaₓ, thetaᵧ, and theta_z:
(d) thetaₓ = arctan(y/z)
(e) thetaᵧ = arctan(x/z)
(f) theta_z = arctan(sqrt(x² + y²) / z)
Use the calculated values of x, y, and z from Step 1 to find the angles thetaₓ, thetaᵧ, and theta_z.
Remember to convert the angles from radians to degrees if desired.
By substituting the given values and solving the equations, you will be able to find the x, y, and z components of the force and the angles thetaₓ, thetaᵧ, and theta_z.