Using a vector diagram, why is it easier to do chin ups when your hands are 30 cm apart instead of 90cm apart?

Question

assume that the force exerted by your arms is the same in both cases

Answer 1

Draw the vector diagram.

 

Your arm muscles have to pull your body up against the force of gravity (pulling you directly down).  If your arms are vertical, all of you power lifts you strait up.  With your arms 30 cm (about 1 foot) apart they pull this way.  Your vector diagram would show this.

 

However, if your arms are 90cm (a little less than 3 feet) apart, they are both pulling together and pulling upward.  Your vector diagram also shows this.  Your arms exert the combined force -- part pulls together and part pulls you upward.

 

And, if your arms are further apart, it gets worse quickly.  The vector diagram is the key to this problem!

Answer 2

It is easier to do chin ups when your hands are 30 cm apart because you are acting against gravity.  Gravity is going to push you down, when you really want to move upward.  Also, your body acts like a weight.  Your arms are not strong enough to defy that weight at such a long distance. When your arms are at a shorter distance, the work is much less.