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The Physics of Gymnastics

By Manyvanh & Ali

"The physics of gymnastics is a complicated, something that seems nearly impossible to understand. When you're watching gymnastics in the Olympics on television, the maneuvers that gymnasts do seem extraordinary. How can a person defy gravity the way gymnasts do? How do the beams and bars support the stress of a gymnast landing on them? Thanks to Sir Isaac Newton, the physics of gymnastics can be broken down." (Maeve Rich)

Yurchenko Vault

Why gymnastics?

To start of, it is important to define and identify exactly what gymnastics is. Gymnastics refers to a sport concerning "physical exercises that develop and demonstrate strength, balance, and agility, especially such exercises performed mostly on special equipment" (Gymnastics). Most girls, especially when they are young, participate in gymnastics. We chose this topic because gymnastics is something that we've both experienced as kids. When we were young, we never gave what we were doing a second thought, besides that we were doing it! But now that were older, we are able to explore the science behind the moves we practiced, which is pretty cool!

Vault

So what does Newton have to do with Gymnastics?

To successfully achieve moves in gymnastics, a gymnasts uses all of Newton's Laws. In this example, we look at the vault. This moves beings by the gymnast running and jumping off a spring board. The gymnast must use a good amount of force to propel themselves off the spring board and into the air. But how do they create this force?

We can look to Newton's Laws for the answer to this question. In this case, we can look specifically at Newton's Second Law of Motion, F=ma (F is force, m is mass, and a is acceleration). Since the gymnast can not change their mass, they must work with other factors to increase the force. To get this greater force, they will have to change their acceleration. The equation for acceleration is a=v/t (a is acceleration, v is velocity, and t is time). We can see that the gymnast can increase their velocity during their initial run (by running faster) to increase velocity, and therefore, their force! This creates a maximum velocity before they reach the spring board.

For example, a gymnast who is 45.4 kg (100 pounds) with a velocity of 3.5 m/s over the duration of 7 seconds would have a force of 22.7 Newtons.

The next step for the gymnast is to push off the spring board. This is where Newton's Third Law of Motion comes in. Newton's Third Law states that every object has an equal but opposite reaction. So when the gymnast pushes off the spring board, the springs in the spring board push back, propelling the gymnast into the air. This is why they needed to create a higher velocity. The greater the force on the spring board, the greater force it will give back, and the higher the spring will propel them up!

After the gymnast pushes off the spring board, they push off the saw horse. This again, is Newton's Third Law. Once the gymnast is in the air for good, they can no longer create any more momentum, and must work with what they have at this point. This is an example of the Law of Conservation of Momentum. This law derives from Newton's First Law of Motion, which states that an object in motion will stay in motion, and an object at rest will stay at rest. Once the gymnast pushes of for the last time before they do their flips and twists, they can no longer create any more momentum, and are stuck with the motion that they are already in. This is why the steps leading to flight are so important in the vault! (The Physics of Gymnastics)

Course Diagram

How does energy work in gymnastics?

As you could probably guess, energy plays a huge part in gymnastics. First we must look at the run the gymnast starts off with. We can calculate the initial energy the gymnast has during the run, which is the maximum they will have throughout. This energy is known as kinetic energy. We can see this in the equation KE= (1/2)mv2 (m is the mass and v is the velocity). By looking at this equation, you can see that the more velocity the gymnast creates by running, the more kinetic energy they will have for the rest of the vault itself.

For example, if the gymnast is the same as stated above, the have a mass of 45.4 kg, and a velocity of 3.5 m/s, then their kinetic energy would be 278.1 joules.

After the run, the gymnast goes into a hurdle step onto the spring board. This kinetic energy is then transferred into the springs of the board as potential energy. The equation that shows this is PEelastic=1/2kx2 (k is the spring constant and x is the displacement of the spring from equilibrium position). If the gymnast creates a great enough force at this point, they can push the board down further, creating a greater displacement of the springs, and therefore, increasing its potential energy.

For example, assuming that no energy is lost due to friction or air resistance, the potential energy would be the same as the kinetic, 278.1 joules. So if the spring compresses 0.15 meters, the spring constant would be 24,720 N/m.

The next step for the gymnast is to push off the saw horse. When the gymnasts pushes, or bounces, off the horse, their arms act as a spring from the horse. In this sense, they further increase the potential energy. This helps the gymnast to create flips and twists in the air, and have a good landing. When looking at the vault, it is also important to note that the gymnast has both translational and rotational kinetic energies, along with gravitational potential energies. The translational and rotational components can be looked at independently, because they do not effect each other.

When the gymnast is in the air, their center of gravity creates a parabolic path in the air, like that of a projectile. Gravities downward acceleration causes the gymnast’s height to decrease, and eventually, to land. At this point in the vault, the gymnast must try to gain both a maximum height and horizontal distance. This is done by changing the angle the initial velocity is directed (when the gymnast leaves the horse). (Gymnast Corner)

Pre-Flight Diagram

How to gymnasts create cool twists and flips?

In order to create the types of twists and flips that make the vault so entertaining to watch, the gymnast must take advantage of two things: torque and angular velocity. To start off, we look at torque. Torque is the amount of force needed to get an object (the gymnast) to rotate. The more torque, the more the gymnast will be able to rotate; and the greater the torque, the greater the angular velocity is. Angular velocity is the speed at which an object rotates. So in essence, the more force put on the gymnast's body, the faster they can rotate. (Physics Behind Gymnastics)

But how can the gymnast create more torque? Due to the Law of Conservation of Momentum, the gymnast can only change their speed to create more twists once in the air. The best way to look at this is by the equation for angular momentum, L=rmv (r is distance from the axis of rotation, m is the mass and v is the velocity). Because the gymnast can't change their momentum or mass, they will have to lower the distant the body is from the axis of rotation (r) to create a higher velocity. In certain types of vaults, the gymnast can do this by tucking their arms and legs in closer to their body during the flips and twists. This will lower the distance from the axis of rotation (through the middle of the gymnast's body). They can then extend their body to slow down when landing or twisting. You can also see this in the equation Iω=Ifωf (I is inertia, and ω is angular velocity). When in the tucked position, the gymnast creates a lower moment of inertia, which in turn creates a greater angular velocity. (Cardinal Scholar, The Physics of Gymnastics)

For example, if you are trying to solve for the gymnasts angular momentum, you could multiply their mass, 45.4 kg, their velocity, 3.5 m/s, and their distance from the axis of rotation, 0.15 m. This would give the gymnast an angular momentum of 23.8 N-m-s.

Flight Diagram

Inertia Diagram