Video Transcript
A particle is in equilibrium under two forces 𝐅 sub one is equal to three 𝐢 plus 𝑥𝐣 newtons and 𝐅 sub two is equal to 𝑦𝐢 plus two 𝐣 newtons. Find the values of 𝑥 and 𝑦.
We’re given two forces 𝐅 sub one and 𝐅 sub two acting on a particle. And we want to find the unknowns 𝑥 and 𝑦 in the direction of unit vectors 𝐣 and 𝐢, respectively. And we’re told that the particle is in equilibrium under the action of these two forces.
Now, recall that the resultant force acting on any body is the sum of all the forces. So, for 𝑛 forces, the resultant force is 𝐅 sub one plus 𝐅 sub two and so on to 𝐅 sub 𝑛. For our two forces then, this is 𝐅 sub one plus 𝐅 sub two. That is three 𝐢 plus 𝑥𝐣 plus 𝑦𝐢 plus two 𝐣. And since to sum vectors in rectangular form we add like for like coefficients, we have three plus 𝑦𝐢 plus 𝑥 plus two 𝐣.
And now recalling that forces are in equilibrium if the resultant force is equal to the zero vector, we know that if a vector is equal to zero, its individual components must be equal to zero. Hence, for our 𝐢-component, we must have three plus 𝑦 is equal to zero. And for our 𝐣-component, we must have 𝑥 plus two is equal to zero. Now, if three plus 𝑦 is equal to zero, subtracting three from both sides gives us 𝑦 is equal to negative three. And if 𝑥 plus two is equal to zero, subtracting two from both sides gives us 𝑥 is equal to negative two.
If a particle is in equilibrium under the two forces 𝐅 sub one and 𝐅 sub two then, 𝑥 is equal to negative two and 𝑦 is negative three.
