Chapter 3: Problem 30

Why do divers executing midair somersaults pull their legs in against their bodies?

Short Answer

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Answer: Divers pull their legs in against their bodies while executing midair somersaults to increase their rotation speed and execute the somersault more efficiently. This is due to the principle of conservation of angular momentum, which states that the angular momentum of an isolated system remains constant if no external torques act on it. When a diver pulls their legs in, their moment of inertia decreases, causing their angular velocity to increase and allowing them to complete the somersault quicker in midair.

Step by step solution

Understand Angular Momentum

Angular momentum (L) is the rotational equivalent of linear momentum and can be defined as the product of an object's moment of inertia (I) and its angular velocity (ω). Mathematically, it is represented as: L = Iω The moment of inertia depends on the mass distribution of the object and its rotation axis, while the angular velocity is a measure of how fast the object is rotating.

Conservation of Angular Momentum

The principle of conservation of angular momentum states that the angular momentum of an isolated system remains constant if no external torques act on it. This can be written as: L_initial = L_final where L_initial is the initial angular momentum and L_final is the final angular momentum.

Diver's Initial Angular Momentum

Initially, during the dive, the diver's body is extended, and they have a larger moment of inertia as their mass is distributed further from their axis of rotation. Let's denote the initial moment of inertia as I1 and the initial angular velocity as ω1. The initial angular momentum L1 would be: L1 = I1ω1

Diver's Final Angular Momentum

When a diver pulls their legs in during a somersault, their mass distribution changes, and they now have a smaller moment of inertia (I2) related to the axis of rotation. However, due to the conservation of angular momentum, their angular velocity (ω2) must increase to keep the angular momentum constant. The final angular momentum L2 would be: L2 = I2ω2

Apply the Conservation of Angular Momentum

According to the conservation of angular momentum principle, the initial and final angular momenta should be equal. Therefore, we can write the conservation of angular momentum equation for the diver as: L1 = L2 I1ω1 = I2ω2

Interpret the Results

Since the moment of inertia I1 is larger than I2, this equation tells us that when a diver pulls their legs in against their body, their angular velocity (ω2) will increase to keep the angular momentum constant. This increase in angular velocity allows the diver to complete the somersault quicker in midair. So, divers pull their legs in against their bodies to increase their rotation speed and execute somersaults more efficiently.

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Why do divers executing midair somersaults pull their legs in against their bodies?

Short Answer

Step by step solution

Understand Angular Momentum

Conservation of Angular Momentum

Diver's Initial Angular Momentum

Diver's Final Angular Momentum

Apply the Conservation of Angular Momentum

Interpret the Results

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