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3. Weight Room Mechanics

The Weight Room Mechanics section is the most difficult part of the Algebra for Athletes program.  In this section you'll learn not only some new math, but also some physics and some things that usually aren't taught until engineering school.  But if you want to see how algebra can be used on the job, this is the best place.

In this section, we're going to see how to use some of the laws we just learned to design weight room equipment. We'll see how to determine the force on a part and then determine how large the part needs to be to support the force.

3.1 Weight Lifting Forces 

There are basically two different types of motion involved in weight machines.  Some machines use linear motion while others use rotation.  Seeing the difference between the two will help explain the mechanics of weight lifting.

3.1.1 Linear Forces

The squat machine helps explain some of the physics of weight lifting.  In the machine in the figure, a bar slides up and down between two sets of guides.  Weights are placed on the ends of the bar.  With this machine, there are generally two forces at work.  Gravity  pulls the weights which creates a downward force .  The other force is generated by the athlete's muscles (mostly the quadriceps ).  To support the weight, the athlete's legs must push up with a force equal to the force of the weights.  The quantity of force, like the force generated by the muscles, can also be measured in pounds.  You would say that to support 100 lb. of weights, the legs must generate 100 lb. of force in the opposite direction.  To move the weights, the athlete's legs must exert a force greater than that of the weights.  Again, in order to explain the algebra, we're going to simplify things.  In all of the problems we look at here, we'll focus only on the amount of force necessary to support the weight rather than to move it.

3.1.2 Rotational Forces

As weight lifting equipment goes, the squat machine in the figure above is unusual.  Most machines donít just move up and down, they rotate.  The figures below show several machines that use rotation to generate the forces.

In these machines, a bar is attached to a hinge.  The athlete's muscles try to rotate the bar in one direction while the weights try to rotate it in the other direction.  Muscles and bones also use rotation to generate forces.  When the muscles contract, the bones rotate about their joints.  Rotational forces or turning forces, like those created by weights or an athlete's muscles are known as moments.  Moments occur in nearly all types of machinery and structures.  In the mechanical world, the moment is often called torque.

Anyone who has ever done a curl before knows that the weight is hardest to move when it is the farthest from your body.  In Figure 3.3, the weight is much harder to move at point B than it is at point A.  This is because the rotational force created by the weight during a curl is the greatest at point B.  The moment created by the weight tries to rotate the forearm in the clockwise direction.  The purpose of curling, of course, is to develop the biceps muscle by resisting the moment created by the weight.  The biceps muscle therefore tries to rotate the forearm in the counterclockwise direction.  During the curl, the forearm rotates about the elbow.  In mechanical terms, the point of rotation is known as the hinge point or axis.   Notice that the weight always pulls the object straight toward the ground.  The line that indicates the direction of the force is known as line of force. 

Figure 3.3 One-arm Curl 

We find the moment, M, by multiplying two factors:

            1. The weight of the object, F (or force); and

            2. The distance between the hinge point and the line of force, l (length). 

The moment, M; the force, F; and the length, l, are related by the following equation:

                                                            M = Fl                                                             (3-1) 

From the diagram, the moment created by the weight at point A is:

                                    M_A = Fl_A= (20 lb.)(5 in.)  =  100 in. lb.

To moment at point B is:

                                    M_B = F_lB= (20 lb.)(12 in.) = 240 in. lb.

This shows that the weight is more than twice as hard to move at point B than it is at point A. 

The distance used to calculate the moment is the perpendiculardistance between the hinge point and line of force.  The perpendicular distance is known as the moment arm.  Since the moment arm is longer at point B, the moment is greater at B.  This is why the weight is harder to curl at point B. Later weíll see how to calculate moments without knowing the perpendicular distance.

The following bench machine problem shows an example of moments.

Example 3.1:  Figure 3.4 shows the side view of the bench press on a universal machine. The bench machine is loaded with 200 lb.  The perpendicular distance from the weights to the hinge point is 48 in.  By pushing up, the athlete applies a force to the machine at the bars.  The distance shown from the hinge to the bar is 60 in.  At balance, the moment created by the weights will equal the moment created by the force of the athlete's muscles.  The clockwise moment created by the weights must equal the counter-clockwise moment created by the lifter.

The moments of the weights and muscles are related by the equation:

        where

 = upward force at the bar, in lb.

                           = length between the bar and hinge, 48 in.

                           = force of the weights = 200 lb.

                           = the length between the weights and hinge point = 60 in.

How much upward force, , is required at the handles to balance the system?  (The weight of the bar itself is ignored in this problem.) 

Solution:   To find the force necessary to balance the system, we need to solve the equation for .  Using the Multiplication Property of Equality, we divide both sides by . 

            Since             = 200 lb.

                           = 48 in.

                           = 60 in.

Then             

An upward force of 160 lb. on the handles is necessary to balance the system.