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Ask a Physicist: Workouts and Wingspan

Do long arms give you a disadvantage at the gym?

Bernie, from San Francisco, wrote in yesterday:
My co-worker and I are having an discussion about weight lifting. The basic question is: does it require more force (strength) to bench press a given mass if your arms are longer? Holding everything else is the same does the increase in distance (longer arms) give that person a disadvantage over a person with shorter arms?
Thanks for writing in! Being a stereotypically scrawny physics-type, I wouldn't have thought of the question, but it's a great one. The short answer to your question is "yes", although there's a bit of terminology we ought to get straight here—the force exerted by each person would be the same, but the work would be different.

In physics, force is defined by Newton's second law: mass multiplied by acceleration, or F = m x a. The same way that the unit of distance is a meter and the unit of time is seconds, the standard unit of force is Newtons. And since Earth's gravity pulls mass down with a constant acceleration (g = 9.81 m/s^2), we can describe an object's weight in Newtons as well—the object's mass times Earth's gravity. A one kilogram object exerts a downward force of 9.81 Newtons, and requires that same force to be lifted. No matter how far you're lifting it, you're still applying 9.81 Newtons of force—even if you're just holding the object steady at a certain height.

Work, on the other hand, is defined as the integral of force with respect to distance. It's important to note that holding an object at a certain height and then moving horizontally with it isn't work in the physics sense—it's only work if you're changing the object's potential energy, i.e. lifting it up against Earth's gravity.

That means that the amount of work done, i.e. the potential energy added to the weights you're bench-pressing, is proportional to your arm's length. A quick measurement revealed my arms to be 63 centimeters long, from armpit to the center of a closed fist, while the arms of my colleague James (who happens to be able to bench 300+ lbs) are 56 centimeters long. That means that, for each rep, James would only be doing about 89% the work I'd do, if we're each lifting to full extension.

"Proportionally," said James, "that means you should be able to bench, what, 270 lbs?"

Yeah, right.

Stephen Skolnick


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