Question

The terminal velocity of fall of lead shots in the cylindrical vessel is influenced by the nearness of the walls of the vessel. Hence the velocity in formula is divided by a factor \([1+\\{(2.4 r) / R\\}] .\) Where \(r\) is the radius of the lead shots and \(\mathrm{R}\) the radius of the vessel. This is called.......... (A) Ladenburg correction (B) Velocity correction (C) End-correction (D) Rydberg correction

Short answer

The correction factor in the terminal velocity formula, \(\frac{1}{1+\{(2.4r)/R\}}\), is called the (A) Ladenburg correction.

Step-by-step solution

Option A: Ladenburg Correction

The Ladenburg correction, which is often applied in the context of fluid mechanics or terminal velocity experiments, accounts for the influence of the wall of the cylindrical vessel on the terminal velocity of particles. Therefore, it seems like a good fit to describe the given correction factor in the terminal velocity formula.

Option B: Velocity Correction

Velocity correction, in general, refers to any modification of the velocity in a certain situation. In this case, the given correction factor is specific to the terminal velocity of lead shots in a cylindrical vessel. While the term "velocity correction" could be applied broadly to any scenario that requires adjusting velocity, it does not specifically relate to the given correction factor.

Option C: End-correction

End-correction, typically used in the context of acoustics or wave propagation, is a correction factor added to compensate for the effect of the end of a tube or pipe on the resonant frequency of an emitted sound. This correction does not seem to have any relevance to the given terminal velocity correction factor in this exercise.

Option D: Rydberg Correction

The Rydberg correction refers to a modification in atomic spectroscopy based on the Rydberg formula, which estimates the energy levels and frequencies of spectral lines. This correction is not related to the terminal velocity of particles or fluid mechanics, and therefore, it does not describe the given correction factor. Based on these definitions, it is clear that the correct term for the given terminal velocity correction factor is: (A) Ladenburg correction

Key concepts

Terminal Velocity

Terminal velocity is a key concept in fluid mechanics. It refers to the constant speed that an object eventually reaches when the force of gravity pulling it downward is balanced by the drag force acting upward. This means the object no longer accelerates and instead moves at a steady rate. You'll often see this phenomenon when objects fall through air or another fluid. For example, a skydiver jumping from a plane accelerates only until the air resistance builds up to counteract the gravitational pull. At this point, the skydiver reaches terminal velocity and maintains a constant speed until deploying a parachute.
In mathematical terms, terminal velocity can be expressed as: \[ v_t = \sqrt{ \frac{2mg}{\rho AC_d}} \] where:

• \(v_t\) is terminal velocity
• \(m\) is the mass of the object
• \(g\) is the acceleration due to gravity
• \(\rho\) is the fluid density
• \(A\) is the cross-sectional area
• \(C_d\) is the drag coefficient

Understanding terminal velocity helps in predicting the behavior of objects moving through fluids, which is essential for many engineering and physics applications.

Ladenburg Correction

The Ladenburg Correction is particularly relevant when considering the terminal velocity of particles in a fluid that is constrained by a cylindrical boundary. This correction addresses how the proximity of the vessel's walls affects the velocity of particles falling through the fluid. Essentially, the Ladenburg Correction modifies the terminal velocity to account for the boundary effects in a confined space, like a cylindrical vessel. The correction factor used is \[ \text{Corrected velocity} = \frac{v}{1 + \left( \frac{2.4r}{R} \right)} \] where:

• \(v\) is the uncorrected terminal velocity
• \(r\) is the radius of the particle
• \(R\) is the radius of the cylindrical vessel

This correction is crucial in laboratory settings when precise measurements of terminal velocity are necessary, and the presence of walls can otherwise skew the results. By applying the Ladenburg Correction, researchers and engineers can ensure that the observed values more accurately reflect the intrinsic properties of the particles themselves, rather than artifacts introduced by their environment.

Cylinder Flow

Cylinder flow in fluid mechanics involves understanding how fluid moves around cylindrical objects. This topic is vital in many industrial and natural settings, like the flow of fluids around pipes or the movement of air around a tower. When a fluid flows past a cylinder, it creates different flow patterns, depending on factors like fluid velocity and viscosity. Here's a basic breakdown of what happens:

• At low speeds, the fluid flows smoothly around the cylinder, creating a symmetric, laminar flow. This is predictable and orderly.
• As the speed increases, the flow becomes turbulent, creating vortices in the fluid wake behind the cylinder. This can cause drag and vibrations.

A famous phenomenon related to cylinder flow is the "Kármán vortex street," which occurs at certain flow velocities and is significant in understanding aerodynamics and structural responses to fluid flows.
An important consideration when studying cylinder flow is the "Reynolds number," \[ Re = \frac{\rho vD}{\mu} \] which helps predict flow patterns by relating fluid density \(\rho\), fluid velocity \(v\), cylinder diameter \(D\), and the fluid's dynamic viscosity \(\mu\). Understanding these concepts enables engineers to design more efficient systems to manage and control fluid flow around cylindrical objects.