Center of mass of motor: rotational equilibrium

Question

Two people place a motor on a $2$m uniform board that weighs $200$ newtons. They lift it, On one side one person lifts with a force of $600$ newtons and on another end the other person lifts it with a force of $400$ newtons. Where is the center of gravity of the motor.

Here is my work: I know that together it took them $1000$ newtons to lift, and I know the board weighs $200$. That leaves $800$ newtons as the weight of the motor. Let $x$ be the distance from the $400$ Newton's end to the center of gravity of the motor. Let the $400$ newton end be the pivot. The sum of the torques must add to $0$ so we have,

$$600(2)-800x-200(1)=0$$

Which suggests that $x=1.25$ meters from the $400$ newton force or $0.75$ meters from the $600$ newton force.

However this is not the answer, here is the answer.

It is problem $8$. I don't really understand their work, but It seems to me there are calculating the center of mass of the whole system instead. Am I right?

Thanks in advance.

Answer 1

If you read the answer carefully, they agree with your calculation when they say

the distance of the motor from the end where the 600 N force is applied is ... 0.75 m

So the (center of gravity of the) motor is 0.75 m from the heavy end.

They then go on to say that the "center of gravity" (of the system... implied but not said explicitly) is 0.80 m from the same end.

Your confusion is with language, not with physics.

Answer 2

You are correct that the question asks for the center of gravity of the motor and the solution gives the center of gravity of the entire system. You can see the position of the motor is given in the second last sentence of the solution and it matches what you found.

This example illustrates a pattern you will become familiar with where book publishers spend more time focusing on churning out new editions of textbooks as quickly as possible and don't seem to care what goes into the books. Hopefully you were able to check the book and solutions out of a library (or something) and you didn't give these people any of your money.