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## Convert pound inch² [lb·in²] to pound-force inch second²

1 pound inch² [lb·in²] = 0.0025900791551159 pound-force inch second²

#### Acceleration

Did you know that fighter pilots wear G-suits that apply pressure to the stomach and legs to prevent blood from rushing away from the brain during acceleration?

The flywheel is built with a high moment of inertia, to ensure that it resists change in the speed of rotation. This allows for the storage of energy, generated by rotating the flywheel.

Overview

In Sports

High Moment of Inertia

Low Moment of Inertia

## Overview

The moment of inertia, also sometimes written as MOI, is the property of a body to resist change in rotational motion. The higher the moment of inertia — the greater this resistance. It is often compared to mass for linear motion because mass indicates how much a body resists change in linear motion. While with linear motion the distribution of weight does not affect this resistance, the distribution of weight along the body that rotates does play a role in the magnitude of the moment of inertia.

This centrifugal governor device uses the moment of inertia to control the speed of an engine by reducing the amount of fuel when the speed reaches a set limit. The engine rotates the two balls above, and as the speed increases, they move apart, increasing the moment of inertia of the device. When the moment of inertia reaches a set value, they reduce the fuel supply.

To calculate the moment of inertia for common geometric shapes with constant density throughout the object, one can use formulas. Calculus is used in more complex calculations. Two objects with the same mass may have a different moment of inertia, depending on the weight distribution within the object. For example, the formula for the moment of inertia, *I*, of a solid ball of uniform density is:

*I* = 2*mr*²/5

Here *m* is the mass and *r* is the radius. If we have two balls of the same weight, one with a radius twice as big as the other one, then the moment of inertia of the bigger ball would be 2²=4 times larger. Here the radius indicates the distance between the center of rotation and the point on an object that is the furthest away from the center of rotation. If we have a cylinder with mass *m* that is equal to the mass of the ball above, and with length *L* that is equal to the radius of the ball above, then its moment of inertia *I* is:

*I* = *mr*²/3

if the cylinder is rotated about its end, and:

*I* = *mr*²/12

if it is rotated about its center (length-wise). The second formula is derived from the first — here the radius from the center of rotation to the furthest point is half of the length of the cylinder, but because this radius is squared, 1/2 *L* (or *r*) becomes 1/4 *L*² (or *r*²). In any case, we can see that even shifting the center of rotation changes the moment of inertia, but this value is also different for two differently shaped objects, a cylinder, and a ball. Moment of inertia affects performance in sports and mechanics, so it is often changed by modifying the mass or length of the object or the human body.

## In Sports

Athletes often increase or decrease the moment of inertia to improve performance. A high moment of inertia helps maintain the current rotational speed, or to keep balance if the rotational speed is zero. On the other hand, a low moment of inertia allows for easier changes in rotational speed, so decreasing the moment of inertia reduces the amount of energy, needed to increase or decrease rotational speed. The moment of inertia is so important, in fact, that some theories suggest matching sports equipment by its moment of inertia if more than one piece of equipment is used in the same sport. This is sometimes done in golf, for example, where some believe that it helps the golfer to improve their swing if all of their clubs have the same moment of inertia. In other sports athletes vary equipment based on its moment of inertia, depending on the effect that they want to achieve, for example depending on how fast they want to swing. They may also choose equipment with a larger moment of inertia to improve their muscle strength, without having to add mass to the equipment. For example, the moment of inertia affects how much speed a baseball club can give to the ball.

### High Moment of Inertia

The surfer extended his arms out to increase the moment of inertia and to maintain his balance on the surfboard.

In some situations it is important to maintain the rotation, despite the forces, acting on the body to prevent this rotation. For example, an acrobat, a gymnast, a dancer, a diver, or a figure-skater may want to maintain the spin at a constant speed for a specific duration of time. In this case, they may want to have a larger moment of inertia. To increase it they can increase the weight of their body. This could be done by holding additional weights that they can discard once they no longer require the high moment of inertia. This is not always a reasonable solution, especially since the weights may fly in the wrong direction and cause damage or injury. Two people can also hold hands to increase the total moment of inertia during a spin, and then let go of each other. This technique is often used in figure skating.

Alternatively, they can increase the radius from the center of rotation to the furthest point on their body that is away from the center of rotation. To do this they may extend their arms or legs out, or even hold long rods, for example.

An athlete, for example, a diver, may need to increase the moment of inertia before entering the water. When the diver is rotating in the air, he waits until he is facing the correct direction for entering the water, and then straightens his arms and legs to stop the rotation and at the same time to increase the moment of inertia. This helps preserve the zero rotational speed, and the athlete enters the water at the correct angle. This technique is also used by skaters, dancers, and gymnasts to land without injury or fall after spinning in the air.

The weight is distributed across a longer barbell to ensure better balance and safety of the athlete. It is always a good idea, however, to exercise with a partner, who can support the athlete if necessary.

As we saw above, a greater moment of inertia means not only maintaining the existing rotational speed for the body in motion, but also maintaining zero speed for the body that is not spinning. This is useful not only if the person wants to maintain the spin, but also if they want to keep balance while not in a spinning motion. For example, acrobats that walk on ropes often carry a long rod to help them stay on the rope, and not “spin off” and fall down.

Weight lifting often employs a moment of inertia for balance as well. The weight is distributed across the long barbell to make it safer for athletes to lift it during bench press exercises. If a smaller object of the same weight, for example, a sandbag or a kettlebell, is lifted instead, then even a slight push at an angle may cause this object to rotate. As a result, the athlete may lose control and drop it. This is why kettlebells are never used for bench press — they can severely damage or even possibly kill the person using them. Even the weights attached to the barbell sometimes pose danger to the athlete’s health — a large amount of mass concentrated at the ends of the barbell may cause the barbell to spin, and hurt the athlete’s wrists. To prevent this, the barbells meant for extra heavy lifting, also known as Olympic barbells, have a mechanism that allows the weight to rotate about the barbell, without causing the entire barbell to rotate with it.

This kettlebell has a very high moment of inertia. Caution needs to be exercised when working with it because it can be very dangerous if the athlete loses control of it. It is a good idea to move it in gentle swings and to keep it as far as possible from other body parts so that it does not hurt the athlete if she loses control.

Work with kettlebells and other similar objects is usually done by increasing the moment of inertia through displacing the center of rotation, often to the person’s body. For example, when swinging a kettlebell, one does not generally use a wrist, or even an elbow as the center of rotation, but instead spins the entire arm or even the whole body, otherwise, it may be very dangerous, as described above.

### Low Moment of Inertia

The skater brings her arms in, towards the torso, to increase her moment of inertia. Once she does, her rotational speed will increase.

In sports, it is often necessary to accelerate or decelerate angular movement by using as little energy as possible. To do this, the athlete either chooses sporting equipment with a lower moment of inertia or decreases the moment of inertia for his or her own body.

In some cases the overall moment of inertia of the athlete’s body is important. In this situation, athletes decrease their moment of inertia by bringing their limbs closer to their bodies during the spin. This helps them spin faster. Such techniques are used in figure skating, diving, dance, and gymnastics. You do not need to practice one of these sports to experience this effect. Simply try spinning on an office chair, with your arms and legs extended out, and then bring them in towards your body — when you do, your rotational speed will increase.

The moment of inertia is important when using sporting or gaming equipment for swings. Equipment with a lower moment of inertia lets the players swing faster, and thus allows them more time to wait and observe the game, before committing to a swing.

In other sports, athletes rotate only parts of their body, for example, an arm, or just a wrist, often while holding equipment, such as a baseball bat or a golf club. In this case, the weight is distributed along the bat or the club in such a way as to minimize the moment of inertia. This is also important for swords, both real and wooden practice ones, as well as for any other sporting equipment that the athlete swings or rotates, such as bowling balls. The moment of inertia also influences how heavy the sporting equipment feels to the athlete — the lower the moment of inertia, the lighter it feels. A lower moment of inertia generally means faster swings, so it allows the athlete to start the swing later. This is useful when playing against an opponent because it makes it more difficult for the opponent to predict the move. This also allows extra time to predict the trajectory of a ball before committing to a swing — useful in tennis and baseball, for example.

It is important to note, however, that when the swinging speed is the same for two bats, one with a lower and one with a higher moment of inertia, the impact at the same speed with the latter will make the ball go faster, even though swinging the bat with a high moment of inertia requires more energy than swinging the bat with the lower moment of inertia. Thus, equipment with a lower moment of inertia is not always the answer — sometimes to achieve better results the athlete cannot rely on equipment with a lower moment of inertia, but has to increase muscle strength and speed instead and use equipment with higher moment of inertia.

Golf club and tennis racket manufacturers display information related to the moment of inertia on the clubs and the rackets, yet baseball bat makers do not provide this information to the consumers. It is unclear why this is the case, but it may be related to the different marketing strategies of the manufacturers. In any case, if this information is not displayed, it is important to try the equipment out at the store, possibly in comparison with other similar equipment, to pick one that suits you best.

References

This article was written by Kateryna Yuri

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### Mechanics

**Mechanics** is the branch of physics, which studies the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment.

### Moment of Inertia Converter

The **moment of inertia** is a measure of an object’s resistance to any change in its state of rotation. It is a property of a distribution of mass in space that measures its resistance to rotational acceleration about an axis as well as its tendency to preserve this rotation. The moment of inertia is the inertia of a rotating body with respect to its rotation. An object that is rotating tends to remain rotating and will continue to do so unless acted upon by an external net torque. The moment of inertia is also called rotational inertia, mass moment of inertia, or polar moment of inertia of mass.

In SI, the moment of inertia is measured in kg·m², imperial/US units are Lbm·ft².

### Using the Moment of Inertia Converter Converter

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