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The spin rate of a tennis ball determines the aerodynamic forces acting on it. In turn, the spin rate is affected by the aerodynamic torque. If the torque depends on flight speed V, density p, viscosity , ball diameter D, angular velocity w, and the fuzz height, hy, find the important dimensionless variables for this case. Use V, p, and D as your scaling (repeating) variables.

The spin rate of a tennis ball determines the aerodynamic forces acting on it. In turn, the spin rate is affected by the aerodynamic torque. If the torque depends on flight speed V, density p, viscosity , ball diameter D, angular velocity w, and the fuzz height, hy, find the important dimensionless variables for this case. Use V, p, and D as your scaling (repeating) variables.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
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The spin rate of a tennis ball determines the aerodynamic forces acting on it. In turn, the spin rate is a§ected
by the aerodynamic torque. If the torque depends on áight speed V , density , viscosity , ball diameter D,
angular velocity !, and the fuzz height, hf , Önd the important dimensionless variables for this case. Use V ,
, and D as your scaling (repeating) variables. 

### Understanding the Dimensional Analysis for a Tennis Ball's Spin Rate #### Introduction The spin rate of a tennis ball determines the aerodynamic forces acting on it. In turn, the spin rate is affected by the aerodynamic torque. To predict and analyze the behavior of a spinning tennis ball, it is essential to determine the key dimensionless variables. #### Problem Statement The torque on a tennis ball is influenced by several factors: flight speed \( V \), air density \( \rho \), viscosity \( \mu \), ball diameter \( D \), angular velocity \( \omega \), and the fuzz height \( h_f \). The goal is to derive the significant dimensionless variables using \( V \), \( \rho \), and \( D \) as scaling variables (also known as repeating variables). #### Key Given Variables - **Flight Speed (\( V \))**: The speed at which the tennis ball travels. - **Air Density (\( \rho \))**: The density of the air surrounding the tennis ball. - **Viscosity (\( \mu \))**: The measure of the fluid's resistance to deformation. - **Ball Diameter (\( D \))**: The diameter of the tennis ball. - **Angular Velocity (\( \omega \))**: The rate of rotation of the tennis ball. - **Fuzz Height (\( h_f \))**: The height of the fuzz on the tennis ball's surface. #### Objective Identify the primary dimensionless groups that characterize the torque's dependence on the aforementioned variables. #### Approach Utilize dimensionless analysis and Buckingham Pi Theorem to derive the dimensionless variables. This involves normalizing the physical quantities by the given repeating variables \( V \), \( \rho \), and \( D \). By organizing these variables correctly and combining them to form dimensionless groups, we can deftly summarize the complex interactions governing the tennis ball's spin dynamics. ### Conclusion The derived dimensionless variables are crucial for modeling and analyzing the aerodynamic forces and resulting torque on a spinning tennis ball. They serve as foundational components in advanced fluid dynamics and sports physics simulations.
Transcribed Image Text:### Understanding the Dimensional Analysis for a Tennis Ball's Spin Rate #### Introduction The spin rate of a tennis ball determines the aerodynamic forces acting on it. In turn, the spin rate is affected by the aerodynamic torque. To predict and analyze the behavior of a spinning tennis ball, it is essential to determine the key dimensionless variables. #### Problem Statement The torque on a tennis ball is influenced by several factors: flight speed \( V \), air density \( \rho \), viscosity \( \mu \), ball diameter \( D \), angular velocity \( \omega \), and the fuzz height \( h_f \). The goal is to derive the significant dimensionless variables using \( V \), \( \rho \), and \( D \) as scaling variables (also known as repeating variables). #### Key Given Variables - **Flight Speed (\( V \))**: The speed at which the tennis ball travels. - **Air Density (\( \rho \))**: The density of the air surrounding the tennis ball. - **Viscosity (\( \mu \))**: The measure of the fluid's resistance to deformation. - **Ball Diameter (\( D \))**: The diameter of the tennis ball. - **Angular Velocity (\( \omega \))**: The rate of rotation of the tennis ball. - **Fuzz Height (\( h_f \))**: The height of the fuzz on the tennis ball's surface. #### Objective Identify the primary dimensionless groups that characterize the torque's dependence on the aforementioned variables. #### Approach Utilize dimensionless analysis and Buckingham Pi Theorem to derive the dimensionless variables. This involves normalizing the physical quantities by the given repeating variables \( V \), \( \rho \), and \( D \). By organizing these variables correctly and combining them to form dimensionless groups, we can deftly summarize the complex interactions governing the tennis ball's spin dynamics. ### Conclusion The derived dimensionless variables are crucial for modeling and analyzing the aerodynamic forces and resulting torque on a spinning tennis ball. They serve as foundational components in advanced fluid dynamics and sports physics simulations.
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