How do you find the resultant vector?

Question

A father and son are trying to remove a tree stump from the yard by pulling on chains attached to its base. The father pulls with 150 pounds of force at a 60° angle to the ground, and the son pulls with 110 pounds of force from 50° on the other side. With what force is the sump being pulled and in what direction? 

Answer 1

Let F be the father's force, S be the son's force.

 

In this problem, you need to combine the two forces(vectors) by their x and y components.

 

Father's x component : F_x = 150cos(60⁰)= 75 lbs.

Son's x component :    S_x = 110cos(50⁰) = 70.7 lbs

 

Remember that these two x components are in opposite direction,

 

so the combined x component R_x =   4.3 lbs, on father's side

 

Now let's look at the y components:

 

Father's y component: F_y = 150sin(60⁰) = 129.9 lbs

 

Son's y components:  S_y = 110sin(50⁰) = 84.3 lbs

 

The combined y component R_y = 129.9 + 84.3 = 214.2, this is in vertical up direction.

 

now the overall combined vector is found by using Pythagorean theorem:

 

R = SQRT(R_x² + R_y²)  =  SQRT(4.3² + 214.2²⁾ = 214.2 lbs

the direction is given by θ = tan^-1(214.2/4.3) = 88.8⁰

 

So the overall force is 214.2 lbs, the direction is 88.8⁰ to the ground from Fathers side.