Calculating Total Thrust Force in Cartesian Components

[ricky_fusion]

Hi, My name Ricky, Indonesia. I am newber in this section. So, I am sorry if my question might be wrong location.

Homework Statement

* The thrust F of each propeller is 1000 lb.
* The thrust directions are described by the angles
α = 10°
β = 5°
γ = 20°
For more detail you can see in : https://ecourses.ou.edu/cgi-bin/ebook.cgi?doc=&topic=st&chap_sec=02.3&page=case_intro

Ask : What is the total thrust (force) on the boat in Cartesian components after the driveshaft was bent?

Homework Equations

Fx = F sinφ cosθ
Fy = F sinφ sinθ
Fz = F cosφ

The Attempt at a Solution

https://ecourses.ou.edu/cgi-bin/ebook.cgi?doc=&topic=st&chap_sec=02.3&page=case_sol

For Normal Force :
F1x = F sinφ1 cosθ1
= 1000 sin90 cos175 = -996.2 lb
F1y = F sinφ1 sinθ1
= 1000 sin90 sin175 = 87.2 lb
F1z = F cosφ1
= 1000 cos90 = 0 lb

Vector F1 can be written in Cartesian notation as

F1 =-996.2i + 87.2j + 0k lb

For Bent Force :
φ' = tan^-1 (tan α cos γ) = 9.41 degree
φ = 90 - φ' = 80.59 degree

θ2 = 200
φ2 = 80.59

F2x = F sinφ2 cosθ2 = -927.0 lb
F2y = F sinφ2 sinθ2 = -337.4 lb
F2z = F cosφ2 = 163.5 lb

Vector F2 can be written in Cartesian notation as

F2 =-927.0i - 337.4j + 163.5k lb

For Total Force :
FT = F1 + F2

= -1923.2i - 250.2j + 163.5k

My question are : (For Bent Force)
1. What is the basic concept to change the degree?
2. Why is the equation/formula should be spherical not just Fx = cos α?

 

[lippyka]

3. Why we should use tan^-1 (tan α cos γ)?Thanks for your response. 1. The basic concept to change the degree is to use the trigonometric functions, such as sine, cosine and tangent. These functions can be used to calculate the angles of a triangle, which is necessary in this case to determine the angle of the thrust force. 2. The equation uses spherical coordinates because it is necessary to calculate the three-dimensional components of the thrust force. By using spherical coordinates, it is possible to calculate the force components in the x, y and z directions. 3. The equation uses tan^-1 (tan α cos γ) to calculate the angle of the thrust force after the driveshaft has been bent. This equation uses the tangent of the original angle (α) and the cosine of the bent angle (γ) to calculate the new angle (φ').

 

[baba89]

Hi Ricky, welcome to the section! Your question is in the right location and I am happy to help you with it. The basic concept to change the degree is to use trigonometric functions such as sine, cosine, and tangent. These functions help us to calculate the magnitude and direction of a vector in Cartesian coordinates. The reason why we use the spherical form of the equation is because it allows us to calculate the thrust force in all three dimensions (x, y, and z). Simply using Fx = cos α would only give us the force in the x-direction and not account for the forces in the y and z directions. Using the spherical form of the equation allows us to calculate the total thrust force in all three dimensions. I hope this helps clarify your questions. Good luck with your calculations!