George and His Leg Press Problem

George wants to know the answer to a specific problem he finds very interesting.  I don't think it's very interesting, but he's the only person who writes to my column, and beggars can't be choosers.

Say you have a frictionless leg press with an angle of theta.  It will look like this; ours arrived in blue.

Now say you put a brick on it.

George wants to know how much the incline helps him push that brick.  I do understand that the sine, cosine, and tangent functions give us ratios for the lengths of sides in right triangles, so that's half the battle.  But there's a missing piece to this puzzle - the normal force.

Here’s one thing I do remember.  The weight of an object does not normally cause it to fall through the floor, and this phenomenon (the fact that the object is, indeed, not falling through the floor) is explained by the existence of the normal force, which the floor pushes back upwards onto the object.

The normal force is green in the picture below.  Real physicists would draw this green line as an upwards arrow, and they would balance this with a downwards arrow representing the weight of the object.  Real physicists would also tell you that the weight of the object is its mass times the force of gravity, but we don't actually need to know that.  All we need to understand is that in this particular case, the normal force is an upwards force that equals the weight of the object, and the weight is a downwards force.

Here's the part I didn't remember.  The normal force is always perpendicular to the surface/floor, no matter what the angle of the surface is.  In fact, the word "normal" means perpendicular!  So the normal force on the leg press looks like this.

The normal force is directed upwards, at the angle perpendicular to theta.  NW.

Okay, now we’re cooking with fire!

We understand that the weight of the object is a force that is directed straight downwards.  It's drawn here in black.  A real physicist would make this line into a downwards arrow, but I have no such temptation because its measurement is very much finite, and this adventure is one of drawing line segments.

The weight represented by the black line can be broken up into two vector components that are oriented to our interests.  They look like this.

The weight, the black vector, has been broken up into two black vectors, and we could even name them.  (The normal force, which is green, is under one of these vectors.)  Let’s call the line opposite the normal force (pointing SE) the “anti-normal force” and the shorter black line (pointing SW) the “parallel to the surface” force.

You may have seen this coming, but the anti-normal force (directed SE) cancels out the normal force (directed NW).  Normal forces are apparently very handy!

What's left is the “parallel to the surface” one.  (A frictional force would be directed opposite to this, in a NE manner, but we're ignoring that).

Our task is to measure that line, and our clues are theta and weight.  Almost everything I can understand, in this world, in this life, is based on a willingness to draw triangles, and this seems to be no exception.  The implication here is that if I cannot prove whether or not a divine being exists with triangles, the chances of my ability to prove such a thing are rather slim.

Here are some similar triangles.

The weight, W, is the hypotenuse of both black triangles.

The “parallel to the surface” force is the one George cares about, so let’s call it G.

The sine of theta is opposite/hypotenuse; that’s just what sine means.  So

sin(θ) = G/W, or
G = sin(θ)*W.

George couldn't remember if that equation used the sine or cosine function, but thanks to the triangles, it's clear as day that cos(θ) = adjacent/hypotenuse = N/W, where N is the normal (or anti-normal) force.

George, you can use a cosine if you're wanting to know how much work your leg press is doing.

When θ = 30°, and George's brick is 400 lbs, he must overcome

sin(30°)*400 lbs = 200 lbs.

On the other hand, the leg press is exerting a normal force of

cos(30°)*400 lbs = 346 lbs.

Three cheers for the leg press!

When θ = 45°, and George's brick is 400 lbs, he must overcome

sin(45°)*400 lbs = 283 lbs.

On the other hand, the leg press is exerting a normal force of

cos(45°)*400 lbs = 283 lbs. 

Nothing too shocking about that; a 45° angle on a right triangle will have equal opposite and adjacent sides.  In this case, both George and the leg press are jolly good fellows.

When θ = 60°, and George's brick is 400 lbs, he must overcome

sin(60°)*400 lbs = 346 lbs.

On the other hand, the leg press is exerting a normal force of

cos(60°)*400 lbs = 200 lbs. 

We could imagine three cheers for George, but it wouldn’t be prudent to shout them aloud, for leg press machines are not manufactured with 60° inclines.

No, they’re manufactured with 45° inclines, and sin(45°) = about .707.  I don't actually know who needs to hear this, but with a frictionless leg press at 45°,

YOU'RE LIFTING SLIGHTLY OVER 70% OF THE WEIGHT STRAIGHT UPWARDS.

Thank you for writing.

Most Sincerely,

Lan from the Dear Lan column I cannot prove exists, with or without triangles