A Simple Model for the Handstand

17 min readAug 8, 2020

Introduction

The physics experience of Amelia Brown, and the curiosity and project management of Erica Dohring brought two strangers together to answer the question: what is the minimum angle that an individual can execute a handstand at? In other words, if θ is defined as the angle created when moving from the hands to the wrists to the arms, what is the minimum angle possible when maintaining a handstand before falling over? If we define 90° to be when your body is straight up and down, on average, the minimum possible angle is only about 87°!

This question can be re-framed into a tipping point problem. The most simplified answer is that, if you treat both the body and hands as rods, the tipping point of a handstand is 90°. This is due to the fact that the only force acting on the body is the force of gravity, and once the body tilts further than this halfway point it will fall.

However, our body is not truly static when we attempt handstands. The muscles in our hands, wrists, and forearms are constantly engaged and provide counter forces that allow us to maintain handstands at angles greater than or less than 90°. What limits the range of angles is an individual’s physical limitation when extending their wrists (bending away from the body), and one’s capacity to exert enough force to maintain that position. The following analysis will estimate that amount of force required to maintain a handstand at various angles.

Torque

An appropriate approach to solve this problem is to balance the torques acting on the body. When two or more torques act on a singular object and entirely balance each other out, then the object will not rotate. Torque is different from force because it considers both the force and the distance from the axis of rotation that the force occurs. Before tackling a torque analysis of a handstand, let’s first examine the analogous yet more familiar behavior of a door on a hinge. [Figure 1]

The equation for torque is τ=rFsinθ

Let’s break down each of these terms — as we will be considering each of them in our analysis.

r = the distance between the axis of rotation and the point where the force is acting

F = the magnitude (size) of the force that is applied to the object

θF = the angle that is created when moving from the axis of rotation to the point where the force acts and then following the direction of force

While doing a handstand, the wrists act as an axis of rotation, therefore a torque-based analysis is appropriate. Putting all of this together, we can now analyze the torques acting on the body. The first, and least complicated of the torques is the torque due to the force of gravity on the body.

Torque due to gravity

Transitioning from the model of the door to the model of the human body is not a huge leap. The doorway (black region in Figure 1) is analogous to an individual’s hands, where the door itself (brown region in Figure 1) is analogous to an individual’s body. In order to differentiate between the hands and the rest of the body, the hands will be referred to as ‘hands’ and the remainder of the body will be referred to as the ‘body’. From now on, the analysis will focus on the forces acting on the body, and how the torques due to the forces on the body come to balance.

**Figure 2** This figure displays how each of the elements of the torque equation are oriented with respect to the hand, wrist and arm. The torque elements considered in this case are all due to the force of gravity.

Here is our equation for the torque due to gravity: τg = r(Fg)sin(θg)

What is r?

Another way to phrase this question is: how far from the axis of rotation does gravity act?

In reality, gravity acts on every bit of the body. However, any object or system of objects can be simplified to a singular point of mass known as the center of mass (COM). The COM essentially averages the position, concentration and value of the mass in every part of the system. We can utilize this information to simplify where gravity acts on the body to a singular location.

Countless studies and high school physics labs have been conducted to measure the COM of a human being, which means that there is plenty of information on the internet which we can pull from. However, it is important to remember that these cases are almost always calculated while the individual’s arms are by their sides. A handstand requires an individual’s arms to be above their head which redistributes the mass of the body. A simple mathematical adjustment can be done to correct this inconsistency. [Figure 3] If we are considering the COM to be the position where gravity acts on the body, then the value for r, is the distance between the axis of rotation and the newly calculated COM of the individual. The specific expression used to calculate an individual’s COM can be found in Table 1.

**Figure 3** Normally, the COM of a human body is measured when an individual’s arms are down by their side. When an individual’s arms are raised above the head, the conventional COM of the human body is shifted upwards. By approximating the locations of the COM for the arms and the body separately, a new COM can be calculated by using the COM equation.

**Table 1** By using average body segment mass percentages, COMs specific to the sex of the individual, and the final equation which was derived in Figure 3, a new COM was calculated and represented in the above table. The body segment mass percentage of an arm was from *de Leva et al., 1996*, and the standard LCOM was taken from https://hypertextbook.com/facts/2006/centerofmass.shtml.

What is Fg?

Fg is the magnitude of the force of gravity. An equation that may be found in the echoes of your high school physics class is F=ma. The force of gravity is equal to the mass of the object multiplied by the acceleration due to gravity: 9.8 m/s². The mass of the human body is simply the value that you read when you step on a scale.

What is θg?

If you recall from before, we previously defined θF as “the angle that is created when moving from the axis of rotation to the point where the force acts and then following the direction of force”. From this definition, our θg will be defined as 90°- θ. [Figure 2] This analysis aims to calculate the amount of force required to maintain a handstand at various angles. Therefore, the variable θ will remain unknown at the moment.

As proposed at the beginning of the article, the tipping point for a handstand is 90° because this is the only angle where the body can stay upright — assuming complete rigidity and no forces due to active muscles in the wrists to balance.

**Figure 4** A schematic to represent the torque due to the force of gravity on the body while attempting a 90° handstand. The Fg force vector is antiparallel to the r vector, therefore θg = 0°.

To check that our analysis is correct, this assumption can be verified in our torque equation for gravity: τ = rFgsinθg = rFgsin(90°- θ) = rFgsin(0) = 0. At θ = 90°, there is no torque acting on the object, as expected. Therefore the body theoretically will not fall due to gravity if it is at 90°. When the body exceeds or falls short of this tipping point, the torque equation becomes non-zero in value, requiring other forces to hold the body up: enter the hand and wrist muscles. Next we will calculate the torques provided by these muscles to counter the torque of gravity at various angles.

Anatomy

While inverted in a handstand, several small adjustments are required in order to stay balanced. What enables us to make these small adjustments are the multitude of complex muscles, tendons, and bones in our hands and wrists. This complexity does prove to be difficult, however, when analyzing the torque due to the hands and wrists. In an effort to simplify the problem, all of the hand muscles were condensed into two different groups: the muscles that cause forces near the fingers and the muscles that cause forces closer to the wrists. See Figure 5 for more details.

**Figure 5** The blue lines represent the outline of the hand/wrist/arm region. The muscles were sorted into one of two groups, digitorum or carpi, based on their approximate termination and insertion points. This figure displays the simplified path that the two groups of muscles tend to follow, as well as the list of muscles that make up each group.

Here are the assumptions that were made:

The muscles can be separated into two groups: “digitorum” and “carpi”. The names do not restrict the types of muscles included in the groups, they simply reflect the classification of the majority of the muscles in each group.
The “digitorum” group includes the muscles: flexor digitorum superficialis, extensor digitorum, flexor digitorum profundus, and flexor pollicis longus.
The “carpi” group includes the muscles: extensor carpi, radialis longus, extensor carpi radialis brevis, flexor carpi radialis, palmaris longus, flexor carpi ulnaris, and extensor carpi ulnaris.
The digitorum group contains muscles that on average originate (start) halfway up the arm, and insert (end) 6/7 of the way down the hand.
The carpi group contains muscles that on average originate halfway up the arm, and insert 1/4 of the way down the hand.

Before proceeding with the details of the torque equations, I will propose a model that will help us to decide what values each of our variables will represent.

As mentioned above in the introduction to torque, it is the individual’s body that experiences the balancing of torques and rotates around the axis of rotation located at the wrist. This model proves to be slightly problematic for several reasons. One particular reason is because the hand muscles originate in the arm (part of the body), yet insert in the hands (not part of the body). Part of the stabilizing force that we want to solve for is not part of the body. This calls for a simplified and approximated model for how the forces work in this situation. In reality, the muscles that travel down your arm and into your hands prevent any unwanted movement by contracting. An equivalent model can be created which treats the forces from the muscles as support structures that are positioned along the imaginary line travelling from the point of termination in the hand to the point of origin in the arm. The reason for this rearrangement is to simplify the behavior of the forces to act at only one location on the body. See Figure 6 for details.

**Figure 6** Presented with the problem that the forces due to hand and wrist muscles act within both the hands and the body, a new model must be developed in order to continue using our general torque expression of rFsinθ. The proposed model pictured in this figure treats the displacement vector (line between the start and end) of the muscle paths as a sort of support structure that will prop up the arm. This model will allow for a singular force to be analyzed at the top of the theoretical support structures for each muscle group. The forces are represented by the dark red and green arrows in the figure above.

Torque due to the force of muscles

Here is our equation for the torque due to the force of muscles: τ = rFsin(θF)

What is r?

As defined above, r is the distance between the axis of rotation and the point where the force is acting. In the proposed model above, the re-oriented force acts upward originating from a point halfway up the arm. Therefore, for both groups of muscles, r=1/2 Larm.

What is F?

The values for the force due to the digitorum muscle group (Fd) and the carpi muscle group (Fc) are the values that we are looking to solve for in this analysis. Therefore, Fd and Fc are our two unknowns.

What is θF?

θF is the angle created while travelling up r and across F. The digitorum and carpi forces will always point in different directions, so a θF will need to be calculated for each of the two forces.

The value of θF will be calculated using trigonometry and will depend on θ. In fact, three different trigonometric situations need to be evaluated for each θF. The first situation is where the wrist creates a 90° angle. [Figure 7] The second and third situations depend on a new variable called θR.

We will define θR as the angle that the wrist makes such that the elbow is directly above the termination point of the muscle in the hand. [Figure 8] The value of θR is dependent only on the dimensions of an individual’s arm and where the muscle group inserts in the hand. Therefore the value of θR will vary depending on which muscle group you are analyzing. Once the value for θR is calculated, the second trigonometric situation will be analyzed for cases when θ is greater than or equal to θR and the third situation will be analyzed for cases when θ is less than θR. [Figure 9, Figure 10] In order to visualize the three situations see the diagrams below:

**Figure 7** One trigonometric situation that can arise when attempting a handstand, is that the arms and wrist create a 90° angle. Pictured above are calculations for the angle θF in this particular situation, where the value of this calculated angle depends on the individual’s measurements for Larm and the insertion point for each muscle group.

**Figure 8** The angle between the hands and wrists that allows for the insertion point of the muscle group to be directly above the origin point of the muscle group, is defined as θR. The angle θR is important to calculate because it marks the angle at which the trigonometry needed to solve for θF changes. The value for θR will depend on the individual’s measurements for Larm and the insertion point for each muscle group.

**Figure 9** The second trigonometric situation that can arise when attempting a handstand, is that the arms and wrist create an angle, θ, greater than or equal to θR. Pictured above are calculations for the angle θF in this particular situation, where the value of this calculated angle depends on θ as well as the individual’s measurements for Larm and the insertion point for each muscle group.

**Figure 10** The third trigonometric situation that can arise when attempting a handstand, is that the arms and wrist create an angle, θ, less than θR. Pictured above are calculations for the angle θF in this particular situation, where the value of this calculated angle depends on θ as well as the individual’s measurements for Larm and the insertion point for each muscle group.

Final torque equation

In order for the body to remain stationary in a handstand, all of the torques must balance. Therefore, the torque due to gravity must equal the sum of the torques due to Fd and Fc. Our final equation is as follows:

τg= τd + τc

LCOMmgsin(90°-θ) = ½ LarmFdsin(θFd)+ ½ LarmFcsin(θFc)

The values for θFd and θFc can be evaluated for different values of θ depending on how it relates to θRd and θRc as specified in Figures 7–10. This equation has two unknown variables in it: Fd and Fc, so we need one more equation that relates the two.

Data Collection

To determine how the two forces relate, two force plates were acquired and several days were spent doing handstands. The two force plates were placed side-by-side and aligned so that while inverted, the region of the knuckles to the tips of the fingers of each hand was on the front plate while the region of the knuckles to the wrists of each hand was on the back plate. The force plates recorded weight throughout the trials, therefore the positioning of the hands allowed for the front plate to read the weight distribution in the front half of the hands while the back plate read the weight distribution in the remaining half of the hands.

The handstands were executed in front of a wall in order to gain support when falling over the natural tipping point, and then attempts were made to controllably leave the wall and maintain several angles around 90°. This method only allowed for short periods of steady inversion (~1–2 seconds), but it was essential to collect data where the body was suspended only by the muscles in the hand and gravity — not by the wall. Additionally, several trials were needed in order to stay entirely rigid and linear. It takes practice to not arch one’s back and maintain a tight form in an attempt to reach angles less than 90°.

From the data, an approximate relationship arose where 40% of the weight distribution fell on the front plate, while the remaining 60% of the weight distribution fell on the back plate. An approximate correlation was made which assumed that 40% of the force was therefore exerted by the digitorum group of muscles and 60% of the force was exerted by the carpi group of muscles. This approximation will fall apart for angles that stray far from 90°, but only angles close to 90° will be needed for a reasonable handstand analysis. From this information, a second equation of Fd=4Fc/6 can be used to solve for the value of each force at different values of θ.

Results

After the data collection, the final system of equations is complete and ready for use in calculating values of force in the carpi and digitorum muscle groups.

LCOMmgsin(90°- θ) = ½ LarmFcsin(θFc)+ ½ LarmFdsin(θFd)

Fd=4Fc/6

A computer program was composed to allow for the streamlining of the calculation process, as the values for θFc and θFd require the user to evaluate whether the angle of the handstand is equal to 90°, or greater than, less than, or equal to the values of θRc and θRd.

The first set of dimensions used to calculate the values of Fc and Fd are presented in Table 2. These dimensions represent the dimensions of an average female in the United States. The calculations for Fc and Fd were executed for values of θ between 90° and 80°. The results of the calculation can be found in Table 3. The calculations were repeated with the average male dimensions from the United States (Table 4) and the results can be found in Table 5. The values for forces presented in Table 3 and Table 5 demonstrate that to maintain an 85° handstand, the average American would need to exert 40–70 lbf (F) or 50–80 lbf (M) for each muscle group. To maintain a handstand just 5° lower, where θ = 80°, the amount of force required doubles. Approximately 100 lbf is needed for the average female body and up to 160 lbf is needed for the average male body (depending on the muscle group).

In order to contextualize how much force is required by the hands, Table 6 lists the amount of gravitational force (weight) that acts on the average female and an average male as well as a force measurement of average palmar pinch strength. By 80°, both sets of data stipulate that the amount of force required by each group of muscles is equivalent to a significant percentage of that person’s weight. In the case of the average female, the carpi group alone requires a force equivalent to 84% of the individual’s weight. [Table 3] Similarly, in the case of the average male, at 80°, the carpi group requires a force equivalent to 83% of the average male weight. [Table 5] When comparing this to palmar pinch strength, the carpi muscle group in the average female data set exceeds the average palmar pinch strength by the angle of θ = 86°, while the digitorum group reaches this limit at the angle of θ = 85°. [Table 3] In the case of the average male, the average palmar pinch strength is exceeded at the angle of θ = 87° for the carpi muscle group and at θ = 85° for the digitorum muscle group. [Table 5]

In conclusion, these results demonstrate how much force is required to maintain a handstand at even 10° beyond the natural tipping point of θ = 90°. However, the data also shows how realistically, the average palm is unable to sustain enough force to tilt beyond even 3° or 4° of the θ = 90° mark.

Tables 7–11 present calculations for Fc and Fd where the dimensions for the average female were further adjusted to see how changing one variable affects the amount of force required. Table 7 demonstrates that a 30% decrease in height results in a 30% decrease in force required to maintain a handstand at a particular angle. Similarly, Table 8 demonstrates that a 30% increase in weight results in a 30% increase in force required to maintain a handstand at a particular angle. From the approximated analysis laid out for this project, these results point to a direct relationship between both the height and weight of an individual and the force required to stay upright.

A less direct relationship can be seen in Table 9, Table 10, and Table 11 where the values for Lhand, Larm, and LCOM were adjusted. In Table 9, an increase in Lhand by 30% led to a 20% decrease in force required to maintain a handstand. In Table 10, a 30% decrease in arm length caused less than 10% increase in force required. In Table 11, the LCOM that was used reflected that of the average male COM instead of the average female COM. The average male COM is shifted more towards the shoulders whereas the average female COM tends to shift more towards the hips. While inverted, this means that the male COM is closer to the ground than the average female COM. The calculations demonstrate that the closer an individual’s COM is to the ground (when inverted), the less force is required to maintain a handstand at various angles.

The results in Tables 7–11 indicate that individuals who are shorter, lighter, have longer hands and arms, and have a more cranial COM will require the least amount of force to maintain a handstand from both the carpi and digitorum muscle groups.

**Table 2** The above values represent the average female dimensions. Average mass and height are from *Centers for Disease Control and Prevention Anthropometric reference data for children and adults, United States, 2011–2014*, Table 3 and Table 9. Average Larm and Lhand values from *NCSU Anthropometric Detailed Data Tables, 4/21/06* p. 18, 39 (Larm = mean on page 39 — mean on page 18). Note that LCOM is calculated assuming that measurements will be made from the ground up while the individual is inverted.

**Table 3** Using the values from Table 2, the amount of force needed from the digitorum and carpi muscle groups was calculated for values of θ ranging from 90° to 80°.

**Table 4** The above values represent the average male dimensions. Average mass and height are from *Centers for Disease Control and Prevention Anthropometric reference data for children and adults, United States, 2011–2014*, Table 5 and Table 11. Average Larm and Lhand values from *NCSU Anthropometric Detailed Data Tables, 4/21/06* p. 18, 39 (Larm = mean on page 39 — mean on page 18). Note that LCOM is calculated assuming that measurements will be made from the ground up while the individual is inverted.

**Table 5** Using the values from Table 4, the amount of force needed from the digitorum and carpi muscle groups was calculated for values of θ ranging from 90° to 80°.

**Table 6** In order to contextualize the data presented in Table 2 and Table 5, the force due to the weight of a 59 kg individual, an 88 kg individual, and average palmar pinch strength are provided. The value for average palmar strength came from Table 5 of *Grip and Pinch Strength: Normative data for adults* by Virgil Mathiowetz et. al.

**Table 7** The amount of force needed from the digitorum and carpi muscle groups was calculated for values of θ ranging from 90° to 80° using the values from Table 2, except that the value for height is decreased by 30%.

**Table 8** The amount of force needed from the digitorum and carpi muscle groups was calculated for values of θ ranging from 90° to 80° using the values from Table 2, except that the value for weight is increased by 30%.

**Table 9** The amount of force needed from the digitorum and carpi muscle groups was calculated for values of θ ranging from 90° to 80° using the values from Table 2, except that the value for Lhand is increased by 30%.

**Table 10** The amount of force needed from the digitorum and carpi muscle groups was calculated for values of θ ranging from 90° to 80° using the values from Table 2, except that the value for Larm is decreased by 30%.

**Table 11** The amount of force needed from the digitorum and carpi muscle groups was calculated for values of θ ranging from 90° to 80° using the values from Table 2, except that LCOM is calculated using the average male expression of .405L instead of the average female expression of .420L. (see Table 1)

Further Analysis

In reality, the body is not a rigid system. Muscle groups other than hand and wrist muscles must be engaged in order to maintain a rigid shape. For example, three other main axes of rotation in our body are the shoulders, back, and hips. The main muscle groups acting at these axes are the shoulder muscles, the muscles of the abdomen, and the muscles in the butt and thighs. Previous analysis has demonstrated that when executing a handstand at angles greater or less than 90°, the force of gravity exerts torque and will cause the body to rotate around an axis of rotation. The process of maintaining a handstand requires forces by the muscle groups mentioned above in order to prevent the remainder of the body from rotating about the shoulders, back, and hips and ultimately falling over due to gravity.

In order to approximate the amount of force needed for these specific muscle groups, all forces due to each of these muscular regions are assumed to act perpendicular to the body. This means that the angle between r and Fmuscle will always be 90°, and the sinθ term will simplify to 1. Further approximations can be found in Figure 11.

Using the dimensions from Table 2, as well as the force expressions from the bottom row of Figure 11, values for each of the forces due to the shoulders, abs, and butt/thighs were calculated and can be found in Table 12. The results of the calculations show that the shoulder muscles require forces that exceed those required by the hand and wrist muscles by almost 10%. The combination of both the digitorum and carpi muscle groups as well as the shoulder, abs, and butt/thigh muscles reaffirm the fact that a strenuous amount of force is required from several muscle groups in one’s body to maintain a handstand even just a few degrees beyond the θ = 90° benchmark.

**Figure 11** The shoulders, back, and hips all present axes of rotation where the body can easily compensate rigidity. The diagrams pictured, as well as the table of values below, specify the locations of the axis of rotation, the force due to the muscles in the area, the COM, and the relative magnitude of the force of gravity on the remaining body. The final expression for the force required by the corresponding muscles can be found in the last row of the table. The values used to determine the mass of the arms, head, legs, and feet were from the body segment data from *de Leva et al. 1996*.

**Table 12** The amount of force needed from the shoulder, abdominal, and butt/thigh muscle groups was calculated for values of θ ranging from 90° to 80° using the values from Table 2.

Final Thoughts

As demonstrated by the data in Tables 3, 5, and 7–11, the forces required to maintain a handstand increase significantly for every degree that one strays from 90°. For an individual with the average dimensions used in this analysis, by 80° your wrist, hand, and shoulder muscles are each exerting forces equal to a significant fraction of the magnitude of your weight. Not to mention that by the first few degrees, the forces required by the hand and wrist muscles exceed the average values for palmar pinch strength. So, what is the maximum possible angle of a handstand? Using the palmar pinch strength as a limiting factor: about 87°. This number can and will change depending on how strong your wrists, shoulders, abs, and thighs are. Hopefully, the data collected from this analysis can help contextualize the answer to this question for your own dimensions and abilities.

How does this information help to increase the quality of your handstands? This analysis asserts that once you figure out how to get up there in the first place, the best way to prepare for maintaining a handstand is strengthening all of your muscles — especially your wrists. This will help so that when you inevitably waver from the ideal angle of 90°, you will have the strength not only to maintain a handstand, but to correct yourself back to the less-strenuous angle of 90°.

This process of analysis was done with several physics-related approximations, seriously oversimplified biomechanics models, and less than desirable gymnastics skills. Therefore, when it comes to analyzing the results of the calculations, please keep these things in mind. Hopefully, this model will be further developed by many others: including but not limited to those who are interested in collecting more data, improving the biomechanical and physical models, or putting their talents in gymnastics to another good use.

August 15, 2020

After publishing this article, Chris Gatti reached out to make us aware of a similarly titled article of his titled “A Simple Model of the Handstand”, linked here if you are interested in further analysis.