Suppose three masses of one kilogram, four kilograms, and six kilograms are located at points whose position vectors are negative six 𝐢 minus 𝐣, two 𝐢 minus nine 𝐣, and seven 𝐢 plus eight 𝐣. Determine the position vector of the center of mass for this system of masses.
In order to answer this question, we can use this formula to find vector 𝐑, which is the position vector of the center of mass. This is calculated by multiplying the mass of each object by its position vector, adding them up, and then multiplying by one over 𝑀, where 𝑀 is the total mass. We may sometimes see this formula given as the sum of all the masses times their position vectors divided by the total mass. These are equivalent.
We can work out 𝑀, the total mass, first. This is equal to one plus four plus six. This is equal to 11 kilograms. We can now substitute the values for the masses and their position vectors into the formula. When we do this, we have the position vector 𝐑 is equal to one over 11, because that’s the total mass. And then within the parentheses, we have the mass of each object times its position vector. That’s one times negative six 𝐢 minus 𝐣 plus four times two 𝐢 minus nine 𝐣 plus six times seven 𝐢 plus eight 𝐣.
In the next stage of simplifying, we can work out each mass times its position vector. This further simplifies to one over 11 times 44𝐢 plus 11𝐣. We can then distribute one over 11 across the parentheses, which gives four 𝐢 plus one 𝐣, or simply four 𝐢 plus 𝐣. Remember that vector 𝐑 is the position vector of the center of mass. And therefore, we can give the answer that for this system of masses, the position vector of the center of mass is four 𝐢 plus 𝐣.