Three forces act on a particle, which moves with constant velocity 𝐯 equals three 𝐢 minus two 𝐣 metres per second. Two of the forces are 𝐅 sub one equals three 𝐢 plus five 𝐣 minus six 𝐤 newtons and 𝐅 sub two which equals four 𝐢 minus seven 𝐣 plus two 𝐤 newtons. Find the third force, 𝐅 sub three.
In this statement, three things really stand out: first, that the velocity of this particle is constant, then, that we’re given 𝐹 sub one in its component form, and finally, we’re given 𝐹 sub two and its components as well. Let’s carry this information over and work to find this third force, 𝐹 sub three.
Alright, so we have our 𝐹 sub one and our 𝐹 sub two. And we want to solve for 𝐹 sub three, the third force acting on this particle that’s moving at a constant velocity. Because our object is moving at a constant velocity and not accelerating, we know that if we add up all three of these forces, 𝐹 one, 𝐹 two, and 𝐹 three, their sum will be zero. Now, how do we know that? Well, think about it in this way: if the sum of these three forces was not zero, then the object would begin to move with an acceleration; it would not have a constant velocity. But we’re told that it does. So that means that if we add up 𝐹 one, 𝐹 two, and 𝐹 three, we must get the result of zero.
So let’s write out 𝐹 one and 𝐹 two as they’re given to us in component form and then include with the assumption that those two forces plus 𝐹 three must be zero. In that way, we can figure out what 𝐹 three is. So here we have our setup, where 𝐹 one and 𝐹 two are written down in component form. And you see there’s a blank space where the components of 𝐹 three will go. We want to solve for those components based on the answer we know: that adding them altogether gives us zero in each component direction.
So let’s start with the 𝑖 direction. If you look at the 𝑖 component of 𝐹 one and 𝐹 two, three plus four is seven, which means that the 𝑖 component of 𝐹 three- in order for that overall component to be zero, the 𝑖 component of 𝐹 three must be negative seven.
Now we move on to the 𝑗 component. So the 𝑗 component of 𝐹 one is positive five, the 𝑗 component of 𝐹 two is negative seven, adding those two together, we have a negative two. Now to counterbalance that so that the total sum is zero, the 𝑗 component of 𝐹 three must be positive two.
And finally, we move on to the 𝑘 component. The 𝑘 component of 𝐹 sub one is negative six; the key component of 𝐹 sub two is positive two. Negative six plus two equals negative four. For the overall sum to equal zero, that means that the 𝑘 component of 𝐹 sub three must be positive four.
So there we have it. The three components of 𝐹 sub three they’re needed to balance out the forces on this particle so that overall it does not accelerate — that it maintains a constant velocity. In summary, 𝐹 sub three equals negative seven 𝑖 plus two 𝑗 plus four 𝑘 newtons.