Levels of Complexity in the Biomechanical Analysis of Secondary Limb Participation During Resistance Exercise

The fundamental premise of resistance exercise is to load a target muscle by applying a resistance source to a limb that is operated by that target muscle, and then contracting that muscle against the opposing resistance.

The simplest example of this is grasping a dumbbell in one’s hand, and performing a “Dumbbell Curl” (aka a “Biceps Curl”). The resistance cannot be held directly by the biceps brachii, so it must be held in the hand which is located at the distal end of the forearm, on the other side of the wrist joint. Thus, the forearm is the “primary lever” of the biceps, because the biceps connects directly to the Ulna of the forearm. The hand (which has some degree of length) is the “secondary lever”, because it’s connected to the forearm (at the wrist), but is not directly connected to the biceps. The hand essentially lengthens the forearm. Thus, it extends the “moment arm” (increases the magnification to the biceps), but only to a small degree.

There are occasions, however, when the secondary lever is longer than the hand, and is therefore much more influential to the primary lever. In the simplest of these cases, the secondary lever is used mainly as a “connector”—a means of holding / applying the resistance, because the primary limb cannot hold it directly. In this scenario, the secondary limb (which is holding the weight) is mostly inconsequential (maintained at a neutral angle). However, it could be more consequential if it is allowed to tilt away from its neutral angle.

Further, there are occasions when the secondary lever (limb) is pressing against a “ground reaction force” (an immovable surface), rather than simply holding a free weight. In such cases, the secondary lever cannot be held in a neutral position, and muscles (one or more) attached to that limb are therefore loaded to the degree of its moment arm — its degree of tilt away from the neutral position.

Adding more complexity to the mechanical analysis, there are occasions when the secondary limb encounters a lateral (sideways, forward or backward) “ground reaction force”—rather than a simple vertical ground reaction force, and this alters the direction of resistance from that which might appear to be obvious.

The analysis is further complicated with the addition of a “brace” from which the person must apply their effort by default—and that is also combined with a lateral ground reaction force.

Analysis is most complicated when an exercise combines all of the above, plus a non-linear resistance source that has as an “arc” trajectory and that trajectory is opposite the limb’s trajectory.

Here’s a more detailed breakdown of the various tiers of complexity.

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Tier 1:     The Neutral Secondary Lever

Examples: Supine Dumbbell Press / One Arm Dumbbell Row

In the image above, the humerus is the primary lever of the pectorals because the pectorals connects directly to the humerus. The forearm is the secondary lever of the pectorals because it does not connect directly to the pectorals, but is connected to the humerus, and is holding the weight.

In this particular application (i.e., the forearm being held vertically), the forearm (i.e., the secondary lever) is only a connector applying the weight to the distal end of the humerus, because the humerus has no other way of “holding” the weight. It requires the hand, which is attached at the distal end of the forearm.

In the top two examples below, the forearm is in the neutral position; it is neither extending nor shortening the “moment arm” of the humerus (as the pectoral lever). It is also not loading the biceps, nor the triceps, nor the external shoulder rotators, nor the internal shoulder rotators. It is acting merely as a “support beam”—holding the weight directly above the end of the distal end of the humerus.

In the lower two images above, I’ve modified the image by eliminating the forearm, and attaching the hand directly to the end of the humerus. I’ve lined up the upper and lower images, and added lines showing the “moment arm” of each (the upper and lower images).

As you can see, the moment arm is essentially the same—with the forearm and without the forearm. The distance between the shoulder joint (i.e., the pivot used by the pectoral muscles) and where the load is applied, is the mostly same. Thus, the magnification is the same—and the load to the pectorals is the same—with or without the forearm.

In the lower image above, I have removed the hands from the ends of the humerus, and attached a chain instead. Now, the weight is hanging below the “elbow” (the distal end of the humerus), instead of being supported above the distal end of the humerus by the forearm. Still, the “moment arm” (the distance between the shoulder joint and the application of the resistance) is the same. Thus, the magnification is the same, and the load to the pectorals would be the same—whether the weight is held above or below the humerus. It’s the moment arm of the humerus that determines load magnification to the pectorals.

This example demonstrates the simplest use of the “secondary lever”. It acts simply as a neutral connector (a “support beam” or a “pendulum”) of the resistance to the primary lever, without increasing nor decreasing the primary limb’s magnification. It also does not involve the muscles (to any significant degree) which actively move the secondary limb (i.e., biceps, triceps, infraspinatus, subscapularis). It is entirely neutral (“zero moment arm”).

To the untrained eye, it might appear as though the triceps are “extending” the elbow (during Supine Dumbbell Press), but the elbows are actually bending and un-bending passively. The elbow “unbends” simply because the distal end of the humerus rises (as the pectorals contract), and it carries the vertical forearm with it. Of course, there is a small amount of biceps and triceps involvement, preventing the forearm from tipping medially (the triceps) or laterally (the biceps). But as long as the forearm remains perfectly vertical (parallel with gravity, like a plumb bomb or a support beam), there is no need for any significant effort from the triceps or the biceps.

Another example of this passive application of a secondary lever is the One Arm Dumbbell Row.

In the two images below, focus your attention on the distance between the green line on the shoulder versus the dotted line, and the distance between the green line on the elbow versus the dotted line.

In the image above-left, notice that the forearm and the upper arm are both vertical and parallel with gravity. Thus, both limbs have, essentially, a “zero” moment arm. Both limbs are neutral. This means that the muscles which operate the upper arm (in this exercise, ostensibly, the “back” muscles) and the muscles that operate the forearm (the biceps and triceps) are neutral—mostly unloaded—in this starting position.

In the image above-right, you can see that the upper arm lever (the humerus) has changed its angle dramatically, and has moved to the horizontal position, and beyond. Notice how much wider the humerus’ “moment arm” is at this stage (the distance between the shoulder and the weight). This indicates that the muscles that cause the humerus to move from the vertical position to the horizontal position (i.e., posterior deltoids, teres major and latissimus) are most loaded. Conversely, the forearm is still mostly vertical, with a near zero “moment arm”, indicating that the muscles which move the elbow joint (the biceps and triceps) are not loaded much, if at all.

Now, imagine that we could replace this person forearm with a chain or rope hanging from his “elbow” (the distal end of his humerus). Then, we attach a weight on the end of the chain or rope. Doing this would not change the dynamics that is occurring at the shoulder joint (i.e., the muscles that are causing his humerus to rise). In other words, like the previous example, the forearm (during One Arm Rows) is simply connecting the distal end of the humerus and the weight. It’s simply acting as a connector, and if the forearm stays perfectly vertical (at the neutral angle), neither the biceps nor the triceps are loaded.

Tier 2:     The Optionally Active Secondary Lever

Examples: Same exercises as above

In the three images below, you can see how the angle of the forearm influences the degree of magnification of the humerus, which is loading the pectorals.

The image above-left shows a vertical (neutral) forearm / secondary lever. The scale shows 1/1 — the same amount of weight on both sides balances the scale, even though the left side has the weight attached to the “hand” (the hole at the distal end of the secondary lever), while the right side has the weight attached to the “elbow” (the distal end of the humerus).

The image above-center shows how tilting the forearm laterally increases the magnification (the moment arm) of the humerus. The scale shows that half as much weight is needed on the left side of the scale, to balance twice as much weight on the other side.

The image above-right shows how tilting the forearm medially decreases the magnification (the moment arm) of the humerus, such that twice as much weight is needed on the left side of the scale, to balance out the weight used on the right side of the scale.

In addition to increasing or decreasing the moment arm of the humerus, tilting the forearm laterally engages / loads the biceps (commensurate to the degree of the tilt), and tilting the forearm medically engages / loads the triceps (commensurate to the degree of the tilt).

Note: Tilting the forearm sagittally (toward the head or toward the feet) would cause the humerus to ROTATE — either“externally” (toward the head) or “internally” (toward the feet). This would load the subscapularis and teres major (internal rotators), or the infraspinatus and teres minor (internal rotators)—respectively—as they try to prevent further humeral rotation caused by the weighted forearm. This would not be a good thing, as it could strain the rotator cuff muscles if a heavy weight is being used.

A similar thing occurs while doing One Arm Dumbbell Row.

If the forearm (the dumbbell, held by the hand at the distal end of the forearm) is tilted toward the head—decreasing the elbow angle—the moment arm of the humerus is decreased. Thus, the magnification of the load to the muscles moving the humerus, decreases. In addition, the biceps becomes loaded, commensurate to the degree of the forearm tilt.

Conversely, if the forearm (the dumbbell, held by the hand at the distal end of the forearm) is tilted toward the buttocks—increasing the elbow angle—the moment arm of the humerus is increased. Thus, the magnification of the load to the muscles moving the humerus, increases. In addition, the triceps becomes loaded, commensurate to the degree of the forearm tilt.

Tier 3:     The Secondary Lever Meets Ground Reaction Force

Examples: Barbell Squats / Parallel Bar Dips

“Ground Reaction Force” is essentially “an equal and opposite resistance” caused by a person’s effort against an immovable object. For example, the starting blocks used by a sprinter in a track meet, “produce” a ground reaction force that is proportional, and in the exact opposite direction, as the diagnonally-down-and-back force delivered by the sprinter against the starting blocks, when he propels himself forward.

Let’s look at an other example, a bit more closely. A man weighing 200 pounds, preparing to Squat with a 200 pound barbell on his shoulders, delivers a vertical downward force against the ground that is equal to 400 pounds. In turn, the ground “pushes back” with an equal and opposite amount of resistance. This is called “ground reaction force.”

Since the direction of force produced by the man’s weight (plus barbell) is vertically down, the direction of the ground reaction force is vertically up. Thus, the “straight upward” direction of resistance—depicted by the “up” arrow in the images below—is the line of force which will be compared against the angle of the limbs, to determine the approximate percentages of load placed on the participating muscles.

In the image above-left, the lower leg (which can be considered the primary quadriceps lever, as well as the potential secondary lever of the gluteus), the femur (which can be considered the primary hip extension / gluteus lever in this scenario), and the torso (which can be considered a potential hip extension lever, as well as the primary spinal extension lever) are all vertical—parallel with the direction of resistance. Therefore, each of them is “neutral”. Each of the three “levers” are acting simply as “support beams” for the weight being held at the top, in this position.

In the image above-right, you can see that the lower leg is not free to be held vertically. It assumes the angle that it must, by default.

“Allowing” the lower leg to tilt away from the vertical position is not a matter of choice, nor a matter of carelessness—as it could be the case in the previous examples (Tiers 1 and 2). In this scenario, the degree of lower leg tilt is determined by the need to maintain balance. In order to prevent falling forward or backward, an equal amount of mass / weight must be maintained in front of the feet, as behind the feet.

A person performing a standard Squat must produce a direction of force that is directly opposite the direction of resistance—or else risk losing his balance and falling. If he “pushes” in a slightly forward-down direction, he’ll fall backward. If he pushes in a slightly backward-down direction, he’ll fall forward.

The obvious objective (i.e., the requirement) of the exercise is to make the body RISE upward from the descended position, and this can only be done by producing a straight downward direction of force. That direction of force causes the femur to encounter the greater “moment arm”, when the person is in the descended position. It is the 90 degree downward rotation of the hip angle (i.e., the femur) that produces most of the upward movement of the torso. This is exactly like the upward rotation of the shoulder joint (the humerus, when doing Supine Dumbbell Press, shown above) which causes the rise of the dumbbell.

However, unlike the Supine Dumbbell Press, the “secondary lever” (the lower leg) cannot be held vertically—in the neutral position—by choice. It must now “double under” the femur—similar to the medial tilt of the forearm during Supine Dumbbell Press. Given that the quadriceps is essentially the “triceps” of the leg, it becomes loaded—just like the triceps becomes loaded when this happens during the Supine Dumbbell Press. This tilt “shortens” the moment arm of the femur, just as the forearm shortens the moment arm of the humerus, when the forearm is tilted medially during Supine Dumbbell Press.

Note: The forward tilt of the torso, during Squats, isometrically loads the erector spinae. Given that the torso is longer than the lower leg, and given that many people tend to tilt their torso forward quite a bit (in order to maintain balance, given their individual femur and tibia lengths), the erector spine could be loaded MORE than the quadriceps.

This is why the degree of quadriceps participation during Squats is similar to the degree triceps participation during a Supine Dumbbell Press, with a medially tilted forearm.

During standard Squats, the moment arm of the lower leg (i.e., the distance between the foot and the knee, with vertically drawn lines), is very short because the lower leg does not tilt very far from from the mostly vertical position. It does not go far beyond point of parallel with gravity (usually between 20 and 30 degrees from vertical). Conversely, the femur usually reaches a horizontal angle (possibly a few degrees less or more)—so it is mostly perpendicular with gravity. This perpendicular angle is theoretically good for the gluteus, but it’s compromised (to a degree) by the doubling-under of the lower leg, which shortens the femur’s moment arm.

Another example of this type of “Secondary Lever Participation” occurs during Parallel Bar Dips.

In the image below (showing Parallel Bar Dips), I’ve marked the hand as “A” (where the resistance is applied), the elbow as “B” (the joint moved by the triceps), and the shoulder as “C” (the joint moved by the anterior deltoids / pectorals). The green lines are vertical because that is the direction of resistance (as demonstrated on the scale in the earlier demonstration). The direction of anatomical force produced by the person doing Parallel Bar Dips is vertical—on the same plain as the direction of resistance—as it must be, given the straight-down pull of gravity. The person’s downward application of force causes his body to rise straight up—opposite the pull of gravity.

Notice the “moment arm” of the forearm—the distance between A and B. Now, refer back to this similar scenario during Supine Dumbbell Press. It’s the exact same physics, except that now (with Parallel Bar Dips) the forearm is below is below the elbow, whereas it was above the elbow when doing Supine Dumbbell Press. Either way, a vertical forearm is “neutral”, and tilting the forearm toward the shoulder loads the triceps—but only to the degree that the forearm tilts away from the neutral position. If it only tilts to small degree, it only loads the triceps a little bit.

The difference with Parallel Bar Dips and with Squats (Tier 3 scenario)—as compared with Supine Dumbbell Press—is that the person does not have the option of causing his secondary lever (his forearm, and his lower leg, respectively) to be in the neutral position. The application of “ground reaction force” (instead of dumbbells) requires a slight tilt away from the vertical position, in order to maintain balance and to produce force that is directly opposite the direction of resistance.

Notice also, during Parallel Bar Dips, the degree of “moment arm” that occurs with the humerus—the distance between A and C. It has a much greater moment arm, just as occurs during a Supine Dumbbell Press. Therefore, the humerus is loading its corresponding muscle(s)—those that move the shoulder joint (the anterior deltoids and pectorals)—significantly more than the forearm is loading its corresponding muscle (the triceps).

Tier 4:  The addition of lateral Ground Reaction Force (instead of vertical ground reaction force)

Examples:  Barbell Bench Press / Chin-Ups / Lunges

In the images below, we see the descended position, and the extended position, of a Barbell Bench Press.

Unlike the situation which occurs when using dumbbells, having the hands gripped onto the barbell prevents the person from being able to move the arms medially — to bring them toward the midline of the body (i.e., The distance between the hands cannot be moved from “wide apart” at the bottom, to “narrow” at the top, when using a barbell). This results in the person being obligated to produce a linear-outward direction of force. This force originates at the shoulder joint, and moves “through” the hand—represented by the white arrows above.

To help prevent the hands from sliding outward (laterally) the barbell is typically “knurled”, and some people also use chalk. If the bar were not knurled and made even more slick with oil, the hands would certainly slide outward—evidence of the linear-outward direction of force produced by the person.

Just like the sprinter’s starting blocks, this lateral direction of muscle force results in an equal and opposite ground reaction force—represented by the red arrows above. As a result, the direction of resistance is no longer vertical, in terms of the analysis of the “moment arms” of the primary and secondary levers, during barbell Bench Press.

Consider the fact that—in the image above-left—that particular angle of forearm (i.e., vertical) would be considered neutral, when using dumbbells. However, now—when using a barbell—the forearm is not neutral because it is not parallel with the direction of resistance (those red arrows). In this scenario, the triceps are called to action even though the forearms are vertical, due to the angle of the forearm relative to those red arrows. In the image above-left, the forearm angle is about 45 degrees relative to the red arrows (i.e., the direction of resistance). This results in “more than zero” load on the triceps, but “less than maximum”—as a percentage of “the available resistance*.

(* The “available resistance” is defined as the weight being used, multiplied by the length of the operating limb, multiplied by the angle of that limb relative to the direction of resistance…using the simplified reference, that a “parallel to resistance” angle is zero, and a “perpendicular to resistance” angle is 100%).

Note: The lateral direction of force being applied against the barbell—by each arm (the right hand pushing diagonally-right, and the left hand pushing diagonally-left), is itself an equal an opposite force. If it were not for the left hand pushing in a left-ward direction, the force of the right hand pushing in a right-ward direction would cause the entire barbell to shift to the right. That does not happen because each hands’ lateral force (in opposite directions) is equal, and thus keeps the bar centered.

Another example of this type of lateral ground reaction force occurs during Chin-Ups, shown below.

The white arrows in the image above-right indicate the direction of force that must be produced by the person. The red arrows indicate the direction of the ground reaction force that is caused as opposition to the muscular force produced by the person.

Since the grip on the chinning bar does not move (just as the sprinter’s starting blocks, and the ground under the person who is Squatting, do not move), the man’s pulling force results in his body rising toward the chinning bar.

Although it might appear that the direction of muscular force is straight-down-vertical (indicated by the green arrows), that’s actually impossible because no force-producing part of the anatomy is situated directly below the hand. Rather, the pulling-force on the Chin-Up bar originates from the shoulder joint. This results in the right hand pulling diagonally-left (toward the midline), and the left hand pulling diagonally-right (toward the midline), against a bar that does not allow the hands to slide “inward” toward the midline of the body.

This “inward” direction of muscular force—in this scenario—is more perpendicular to the forearms (i.e., the secondary lever) than would be the case if the muscular force could be straight-down vertical (parallel with the forearm). This is the reason why there is a fair amount of biceps participation during Chin-Ups, whereas there is little or no biceps participation during One Arm Dumbbell Rows.

Similar to what occurs during the Barbell Bench Press, the inward pulling of each arm during Chin-Ups opposes that of the contralateral arm. The right arm’s effort of pulling diagonally-left would tend to shift the entire body toward the right, if not for left arm’s effort which is pulling the body toward the left. The equal and opposite diagonal pull of each arm keeps the body in the center.

Notice that any lateral ground reaction force—a sideways force that is separate from the vertical force produced by simple gravity—requires a secondary point of contact, which causes that lateral force. For example, a One Arm Chin-Up (shown below) would produce no lateral ground reaction force (assuming the person is strong enough to do it). The force production required by the person would then only be vertical. However, when the second arm (the second point of contact) is included in the Chin-Up, each arm produces the lateral ground reaction force that opposes the contralateral arm.

Illustrated below is the third example of this type of ground reaction force—Lunges. In this case, it is momentum (initiated by the rear foot, the second point of contact) that produces the lateral force, which then combines with the straight down direction of gravity, and produces a “composite” direction of resistance.

The green arrow in the image above-right indicates the trajectory of the body weight caused by the forward step. This altered direction of resistance (i.e., the vertical pull of gravity combined with the slight forward momentum), is then suddenly halted when the forward foot meets the ground. The angle of “landing” causes an equal and opposite ground reaction force (i.e., the black arrow), which crosses the tibia more perpendicularly than would occur without that forward momentum (initiated from the rear leg). Thus, this ground reaction force (eccentrically) loads the quadriceps more—as a percentage of the weight being used—than would occur with a straight Squat.

Note: When the forward-stepping foot “lands” (makes contact with the ground), you can often feel your foot slide forward into the front part of your shoe, demonstrating the forward momentum and corresponding ground reaction force, which halts your continue forward motion. If the ground were slick and oiled, you would discover (“the hard way”) that this forward momentum is occurring, if you didn’t anticipate it.

If the second half of this Forward Lunge involves returning to the original position, it would require a (concentric) forward thrust from the tibia / quadriceps, such that it would propel the body BACK over the rear foot. That force would cross the tibia more, and engage the quadriceps more, as compared with the person continuing the step FORWARD, out of the Lunge. Stepping forward would engage the gluteus more, because the ground reaction force would then be more perpendicular to the front femur.

Note: Lateral ground reaction force is sometimes incorrectly referred to as “friction force”, possibly because the term “ground reaction force” is thought to pertain only to resistance from the ground. However, “friction force”—as the name implies—is the friction that is created by something rubbing, sliding or dragging across a surface, like a baseball player sliding into home base. If there is no “rubbing, dragging or sliding, there is no friction. If the pushing or pulling is against an immovable “grip” or surface, then it is simply an “equal and opposite direction and amount of force”, which is the definition of “ground reaction force.”

Tier 5:     The addition of a “brace”, combined with lateral ground reaction force

Examples: Swiss Ball Wall Squats / Hack Squat Machine

In the image below, you can see that this person is leaning back against a Swiss Ball, which is against the wall.  This is a “brace” (a second point of contact), and it alters the direction of resistance used during an exercise.

As you can see, her feet–which are placed on the ground in front of her (i.e., not directly beneath her torso and hips)–can only push in a diagonally forward-down direction (indicated by the red arrows), due to leaning back against the brace. If the ground were slick and oiled, her feet would slide forward (indicated by that green arrow). However, since the ground is not slick and her feet are “gripping” the ground, a reaction force is created that is in a diagonally-up-and-back direction of resistance (opposite those red arrows).

Note that, in the image above-right, you can see that her lower legs are vertical. However, they are not “neutral” in this situation, because the direction of resistance is not vertical. Thus, this diagonal direction of resistance loads the quadriceps more, as compared with a vertical direction of resistance. During this exercise, the quadriceps are able to interact fairly perpendicularly (more productively) with the modified direction of resistance.

It’s important to note, however, that the direction of resistance is still not “maximized”. The resistance is not being applied to the lower leg lever 100% perpendicularly. The angle of the resistance, relative to her tibia (in the descended position) is about half way between zero (neutral) and maximum (100% perpendicular). Therefore, the load to the quadriceps is a percentage* thereof.

* Note: Trigonometry would be required to calculate the exact amount of load a muscle would encounter given a particular limb angle relative to the direction of resistance. However, for the sake of simplicity—for the purpose of allowing a general understanding of how a limb angle (relative to the direction of resistance) loads a muscle more, and less—it’s convenient to consider a limb angle that is 45 degrees to the direction of resistance (i.e., half way between zero magnification and 100% percent magnification), a “50% lever”. Thus, it would load its corresponding muscle with approximately half as much load as would occur if that limb were completely perpendicular to the direction of resistance. The objective is to have a sense of how muscle load increases, or decreases it, based on the angle of the operating limb, relative to the direction of resistance.

Also note that a percentage of the “weight” (her bodyweight plus the weight she is holding) is “resting” against the Swiss Ball (i.e., “the brace”). This reduces the amount of “weight” that’s loading the exercise, to some degree. Therefore, calculating the amount of load her quadriceps encounter during this exercise would require an estimation which includes that reduction, plus the “moment arm” of her lower leg angle (i.e., the degree of tilt between neutral and perpendicular), plus the length of her lower leg lever (tibia).

Also, notice the angle of her femur (in the descended position) in relation to the red arrow. The two black lines I’ve placed there show the moment arm of the tibia, but that “space” (between the two lines) is the same as that of the femur. What this shows is that the lower leg essentially negates the moment arm of the femur.

Imagine doing a Supine Dumbbell Press (as shown above), but–instead of medially tilting your forearm a few degrees–you tilt your forearm to the point where the dumbbell is directly over your shoulder (in the descended position). The moment arm of the humerus would then be no greater than that of the forearm. That would cause the triceps to be very loaded, but would essentially eliminate the moment arm (the leverage) of the humerus — so there would be little (if any) pectoral load. This is why attempting to work two muscles at one time is inherently problematic. Either both muscles (both operating limbs) experience a reduced load, or muscle / limb is very loaded while the other one is not loaded much at all.

Thus, the exercise above requires very little dynamic hip extension (i.e., force produced by the glutes, etc.). Rather, it’s the lower leg levers—and subsequently the quadriceps—that are doing most of the work. The knees forcefully extend (from the descended position), which pushes the pelvis/lower spine up-and-back, which causes the ball to roll upward, which carries the torso upward. There is very little moment arm with which the femur can produce force, by way of the gluteus.

In the example below—the Hack Squat Machine—we have a very similar situation as above, except that there are two braces: one on which his back rests, and one which is over his shoulders. Again, his quadriceps produce most of the force necessary to rise out of the descended position. His direction of force (indicated by the red arrow) originates at the pelvis, which leaves the femur with very little moment arm / leverage. That’s what makes this exercise “very good” for the quadriceps.

It’s worth noting that the angle of the sled (apparently 40 degrees, or so) reduces the actual amount of weight with which the exercise is loaded—as a percentage of the total weight (i.e., the sled plus the weights attached to it). If the sled were perfectly vertical, 100% of the weight would load the exercise.

As reference, imagine the sled angle being nearly 100% horizontal. There would be very little (if any) load against which the legs would have to work, even if the sled was loaded with a considerable amount of weight.

During this type of Hack Squat, the erector spinae (the muscles that extend the spine) are entirely without load—which is good, in this application. The weight of the torso is essentially resting on the back-pad, relieving the erector spinae from having to work against a forward pulling resistance (which occurs during standard Barbell Squats).

The “downward” thrust of his legs originates from his pelvis/base of his spine, and pushes through his feet (indicated by the red arrow). That direction of force produces an equal and opposite ground reaction force (the green arrow). That “line of force” is approximately 45 degrees to the lower leg levers, but it also mostly negates the moment arm of the femur, as the operating lever of the hip extension muscles (i.e., the gluteus, etc.).

Again, you can see the inverse relationship between a maximized moment for the quadriceps, and a minimized moment arm for the gluteus. This is always an issue when dealing an exercise that involves two disparate joint functions, and one direction of resistance.

If a person places his feet farther out in front (during Hack Squats), he’ll produce more forward thrust (against the “grip” of his feet on the platform), rather than a downward thrust. However—in this case—either foot position will produce a ground reaction force that loads mostly the quadriceps, and minimizes the gluteus (hip extension). This is because in both cases, the lower leg lever minimizes the moment arm of the femur.

In a sense, you could say that the muscle loading which occurs during a Hack Squat (machine) resembles the muscle loading which occurs during a Sissy Squat, more so than it resembles the muscle loading which occurs during a standard Barbell Squat.

During Hack Squats and Sissy Squats, the quadriceps are predominantly loaded, and the gluteus participation is minimized. During Barbell Squats, the gluteus is loaded more, and there is less quadriceps participation. This can be identified by the examining the moment arm of either lever (the femur and the tibia) in these exercises.

The Hack Squat machine allows more weight to be moved—as compared with Sissy Squats—for several reasons. There is a percentage of load reduction caused by the angle of the sled of the Hack Squat; the lower leg lever usually does not reach the fully perpendicular angle with the direction of the ground reaction force; and the femur angle has a small amount of leverage / moment arm (…there is bit more gluteus participation than occurs during Sissy Squats); and the issue of balance / coordination is eliminated. However, this does not necessarily mean that the quadriceps are more loaded with Hack Squats, as compared with a Sissy Squat. Maximum quadriceps effort is maximum quadriceps effort, either way.

There may also be a psychological component that occurs during Hack Squats, such that the person is compelled to put forth greater effort and perform more reps, given the heightened sense of stability.

Tier 6: The addition of a non-linear (curved) primary direction of resistance, combined with a lateral ground reaction force, plus a brace.

Example: Pendulum Squat

In the image above-left, I’ve placed a large curved green arrow, indicating the trajectory of the lever arm of this machine—a Pendulum Squat. However, that is not necessarily the direction of resistance experienced by the person, because he has a brace (second point of contact) against his back which allows him to push more backward than upward. This, combined with is foot placement—slightly in front of him, rather than directly under him (i.e., in line with his torso)—causes a modified direction of resistance. This is a type of ground reaction force that is similar to the Swiss Ball Squats.

The actual direction of resistance (due to the ground reaction force) is represented by the orange arrow, which is directly opposite the person’s direction of force (lime green arrow), pushing back against the pad. The direction of force originates at his pelvis, and pushes through his feet. That is the line of force, highlighted by the red line at this foot. The other red line (at his knee) allows you to see the moment arm of his tibia (highlighted with a thin white line).

This is a fairly good angle (moment arm) for the tibia, relative to the direction of resistance. The tibia is not  “maximized”—it does not reach a fully perpendicular angle with the line of force—but it is still mostly good. It’s a bit closer to perpendicular with the line of force, than it is parallel with the line of force.

The advantage of this scenario, is that there is essentially no consequence to adding additional weight to the machine, to compensate for the slightly less than perpendicular angle of the tibia, relative to the direction of resistance. His back / spine is totally supported (un-weighted), so there is no risk of excessively forward loading the torso / erector spinae—which occurs during standard Barbell Squats.

Again—similar to what occurs when doing Hack Squats—the angle of the lower leg (which is advantageous for the quadriceps), essentially negates the moment arm of the femur, in terms of gluteus loading. As such, there is much less gluteus loading with this exercise, as compared with quadriceps loading.

More interesting, perhaps, is the fact that the clockwise rotation of the machine’s lever arm, matches the clockwise rotation of the tibia around the knee axis. In essence, the curved downward trajectory of the machine “pushes” the carriage (with the person on it) toward his heels, thereby ensuring a perpendicular angle of force to the tibia.

Ironically, the clockwise rotation of the machine is opposite the counter-clockwise rotation of the femur, around the hip axis. This further minimizes the mechanics that would favor gluteus loading. As the machine rises, it prevents the femur from fully extending—ultimately stopping 40 to 50 degrees short of full gluteus contraction.

In the image below—the final position of the Pendulum Squat—you can see that the knees are fully extended, but the hip joints are not. The knees have traveled approximately 110 degrees, with a fairly perpendicular tibia (relative to the line of force), while the hip joint has only traveled approximately 70 degrees, with femur that had a drastically reduced moment arm (caused by its secondary lever—the tibia).

Also, similar to the Hack Squat, the stability provided by the Pendulum Squat machine, combined with the back support, allows the person to add additional weight without much concern for balance, or risk of injury. This might psychologically incentivize the person to use more effort, as compared with an exercise that does not provide this type of “security” / stability.

Another aspect that is unique to this exercise is that the resistance varies, depending on the location of where, along the curved trajectory, the person and carriage are. When in the fully descended position, the resistance is “least”—because the moment arm of the pendulum has decreased. When in the fully ascended position, the resistance is “most”—because the pendulum has moved laterally farther away from the pivot (as measured with vertical lines), maximizing the machine’s moment arm. This could be viewed as potentially beneficial, because the moment arm of the person’s tibia is maximized in the descended position (where the machine provides a bit less resistance), and his tibia’s moment arm is reduced in the ascended position (where the machine’s resistance is maximized). However, in practical terms, it would likely not result in a significant advantage, as compared with a more “standard” resistance curve.

The exercise below has a very similar biomechanical profile to the Pendulum Squat. The principles are the same—curved trajectory of resistance, back support (brace) allowing the person to push “back” against the brace, creating a ground reaction force that is more perpendicular to the tibia than a standard Squat. But, here again, the mechanics that favors the quadriceps (i.e., the tibia), compromises the gluteus (i.e., the femur). It’s impossible to optimize both at the same time, because they each “need” (would most benefit from) their own, separate direction of resistance.

The disadvantage of this exercise (above)—as compared with the Pendulum Squat—is that it’s more difficult to load (requires hand-holding the weight), and it’s cumbersome to set up (connecting the straps while also picking up the weights). Once in place, however, it’s a good quadriceps exercise.

The above “6 Tiers” illustrate that most exercises have their own, relatively unique biomechanical profile. Yet, each exercise can be dissected and evaluated once the parameters for evaluation are understood.

To an untrained eye, some exercises might seem to have an “obvious” biomechanical profile (i.e., which muscles are loaded seems “obvious”). However, upon closer inspection, there’s more to it than meets the eye. For example, in this exercise below—Sissy Squat Bench—it appears very similar to a standard Squat, but in fact it’s very different.

There are two braces in play here, and they alter the direction of resistance from vertical-down, to ROTATIONAL—pivoting around the brace that is positioned on the posterior side of the proximal tibia (i.e., upper / back side of the lower leg).

In fact, you can see (in this image above) that his feet are not even touching the floor plate—as would occur if the direction of resistance were straight-down vertical (like standard Barbell Squats). In this case, his bodyweight would essentially fall backward, if not for the forward brace in front of his ankle. This rotational force and brace produce an equal and opposite ground reaction force against the front (distal end) of his tibia, which loads his quadriceps.

In the image below-left, you can see that his knee is the axis of rotation—caused by the brace behind the knee, and the shift of his bodyweight posterior to the apex above the knee. This causes a forward thrust of the ankle against the roller–similar to the ground reaction force discussed in Tier 3 (the brace as an additional point of contact, which redirects the resistance).

On a separate note, he is leaning his torso (from this perspective) to the left of (anterior to) the apex above his hip joint (i.e., the base of his torso). This forward leaning “shortens” (reduces the moment arm of) his femur. This is similar to the medial tilting of the forearm during Supine Dumbbell Press, in Tier 2 (the “optional tilt of the secondary lever”).

In the image above-right, the man is allowing his torso to lean all the way back, which is similar to the extreme lateral tilting of the forearm during Supine Dumbbell Press, in Tier 2. It dramatically lengthens the femur, which greatly increases its load magnification to the quadriceps. Notice the two vertical dotted lines, which show the moment arm, and compare that “width” to the moment arm of the image above-left (from his head to his knee).

In the image above-right, there can be no load whatsoever to the gluteus, when the torso is leaning to the posterior side of the apex above the pelvis. The image above-right also has considerable engagement from the rectus abdominis and hip flexors, which are preventing his torso from collapsing backward. The image above-left has a very small amount of gluteal engagement, and a reduced quadriceps load, due to his forward leaning torso.

Identifying and understanding the effects of moment arm changes, and the corresponding increases or decreases in muscle load which are influenced by the angle of a limb relative to the direction of resistance, is critically important to understanding which muscles are loaded, and to what degree they are loaded. Also important is the ability to identify changes in the direction of resistance which are caused by the addition of lateral ground reaction forces, braces, and non-linear trajectories of resistance.

Doug Brignole is the author of “The Physics of Resistance Exercise”, a biomechanics researcher, and a former bodybuilding champion with a 43 year history of competition.

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