A 0.5-kg ball having a diameter of 50 mm is thrown with a speed of 10 m / s and has an angular

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A 0.5-kg ball having a diameter of 50 mm is thrown with a...

A $0.5-\mathrm{kg}$ ball having a diameter of $50 \mathrm{~mm}$ is thrown with a speed of $10 \mathrm{~m} / \mathrm{s}$ and has an angular velocity of $400 \mathrm{rad} / \mathrm{s}$. Determine its horizontal deviation $d$ from striking a target a distance of $10 \mathrm{~m}$ away. Take $\rho_{a}=1.20 \mathrm{~kg} / \mathrm{m}^{3}$ and $\nu_{a}=15.0\left(10^{-6}\right) \mathrm{m}^{2} / \mathrm{s}$, and use Fig. $11-50 .$ Neglect the effect on lift caused by the vertical component of velocity.

Key Concepts

Spin Parameter (Lift Coefficient Dependency on Spin Ratio)

The spin parameter, often expressed as a ratio of rotational speed to translational speed, is used to evaluate how spin affects the lift force experienced by an object in flight. This parameter, along with experimentally derived lift coefficients, helps quantify the Magnus force acting on a spinning projectile, enabling the prediction of lateral deviations from its expected path.

Projectile Motion

Projectile motion is the study of objects thrown or projected into the air, subject only to the acceleration of gravity. It involves analyzing the object's trajectory, time of flight, and range, with adjustments made when additional forces, such as drag or spin-induced effects, act on the object. This concept is foundational for understanding how objects move when launched at an angle in a gravitational field.

Aerodynamic Drag

Aerodynamic drag is the force that opposes an object’s motion through a fluid (such as air). It depends on factors like the object's shape, size, velocity, and the fluid’s density and viscosity. In projectile problems, drag reduces the horizontal and vertical components of velocity, thereby affecting the range and trajectory of the projectile.

Magnus Effect

The Magnus effect is the phenomenon where a spinning object moving through a fluid experiences a force perpendicular to its direction of motion. This force, arising due to differences in pressure around the object, can cause a deviation in the object’s path. In sports and engineering, the Magnus effect is crucial for predicting and analyzing the curved trajectories of spinning balls or projectiles.

Reynolds Number

Reynolds number is a dimensionless quantity used in fluid dynamics to predict the flow pattern around an object by comparing inertial forces to viscous forces. It helps determine whether the flow will be laminar or turbulent, which in turn influences the drag coefficient and, consequently, the aerodynamic forces such as drag and lift acting on the projectile.

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