What is the primary difference between laminar flow and turbulent flow, and why is it important for Stokes' Law?

Laminar flow involves smooth, parallel layers of fluid moving without mixing, while turbulent flow is characterized by chaotic, mixing fluid layers. Stokes' Law is only valid under laminar flow conditions because its derivation assumes this orderly movement. If the flow becomes turbulent, the drag force calculation using Stokes' Law will be inaccurate.

Distinguish between systematic and random errors in the context of the falling sphere viscosity experiment.

Systematic errors are consistent, repeatable errors that affect all measurements in the same way, such as an incorrectly calibrated ruler or starting the timer before terminal velocity is reached. Random errors are unpredictable variations that occur due to chance, like slight inaccuracies in timing or the ball not falling perfectly centrally. Systematic errors affect accuracy, while random errors affect precision.

How does the role of upthrust differ from viscous drag in opposing the motion of a falling sphere?

Upthrust is a buoyant force caused by the displaced fluid, acting upwards regardless of motion, and depends on the volume of the sphere and fluid density. Viscous drag is a frictional force that directly opposes the sphere's motion through the fluid, acting upwards only when the sphere is moving, and depends on the sphere's velocity, radius, and fluid viscosity.

What common error occurs if the ball bearing does not reach terminal velocity before timing begins?

If timing begins before the ball bearing reaches terminal velocity, the measured velocity will be an average of an accelerating phase and a constant velocity phase, which will be lower than the true terminal velocity. This systematic error would lead to an overestimation of the fluid's viscosity when calculated using the derived formula.

What is a common mistake related to the densities used in the viscosity calculation formula?

A common mistake is confusing the density of the sphere ($\rho_s$) with the density of the fluid ($\rho_f$) or using the wrong one in the $(\rho_s - \rho_f)$ term. This error will lead to a significantly incorrect viscosity value, potentially even a negative one if the fluid density is mistakenly subtracted from the sphere density in the wrong order.

What error might arise if the measuring cylinder's diameter is too small relative to the ball bearing?

If the cylinder's diameter is too small, the proximity of the walls can significantly affect the fluid flow patterns around the falling sphere, leading to increased drag (wall effects). This can cause the measured terminal velocity to be lower than it would be in an unbounded fluid, resulting in an overestimation of the fluid's viscosity.

Define the term 'viscosity' in the context of fluid dynamics.

Viscosity is a measure of a fluid's internal resistance to flow. It quantifies the fluid's 'thickness' or its resistance to shear stress. High viscosity fluids flow slowly, while low viscosity fluids flow easily, due to differences in their molecular interactions.

What is terminal velocity and what condition defines it?

Terminal velocity is the constant speed that a freely falling object eventually achieves when the sum of the drag force and upthrust equals the downward gravitational force (weight). At this point, the net force on the object is zero, and its acceleration ceases.

State Stokes' Law and explain the meaning of each variable.

Stokes' Law states that the viscous drag force ($F_d$) on a small sphere moving slowly through a viscous fluid is $F_d = 6\pi \eta r v$. Here, $\eta$ is the coefficient of viscosity, $r$ is the radius of the sphere, and $v$ is its velocity. It describes the resistive force due to fluid friction.

What is the derived formula for the coefficient of viscosity ($\eta$) in the falling sphere experiment?

The derived formula for the coefficient of viscosity is $\eta = \frac{2r^2 g (\rho_s - \rho_f)}{9v_{term}}$. In this equation, $r$ is the sphere's radius, $g$ is acceleration due to gravity, $\rho_s$ is the sphere's density, $\rho_f$ is the fluid's density, and $v_{term}$ is the terminal velocity of the sphere.

Revision Notes

International A-Level

Pearson Edexcel

Physics

1. Mechanics & Materials

Core Practical 2: Investigating Viscosity

Core Practical: Investigating Liquid Viscosity

Summary

This core practical provides a hands-on method to determine the coefficient of viscosity of a liquid by observing the terminal velocity of small spherical objects falling through it. The experiment relies on balancing the gravitational weight, buoyant upthrust, and viscous drag forces acting on the sphere at terminal velocity, allowing for the calculation of viscosity using a derived formula based on Stokes' Law.

1. Definition & Core Concepts

Viscosity is a measure of a fluid's resistance to flow, often conceptualized as its 'thickness'. A fluid with high viscosity flows slowly, while a fluid with low viscosity flows easily. This property is crucial in many industrial and natural processes.
The coefficient of viscosity ( $\eta$ ) is a quantitative measure of this resistance, representing the internal friction within the fluid. It is typically measured in Pascal-seconds (Pa s) or Newton-seconds per square meter (N s m $^{-2}$ ).
Terminal velocity ( $v_{term}$ ) is the constant speed that a freely falling object eventually reaches when the resistance of the medium through which it is falling prevents further acceleration. At this point, the net force on the object is zero.
The aim of this practical is to calculate the coefficient of viscosity of a liquid by measuring the terminal velocity of small spherical objects of known properties as they fall through the liquid. This provides a direct experimental determination of a fundamental fluid property.

2. Underlying Principles: Force Balance

3. Experimental Methodology

The equipment typically includes a long measuring cylinder, the viscous liquid to be tested, a stand and clamp, a metre rule, rubber bands, steel ball bearings of various sizes, digital scales, Vernier calipers, and a digital stopwatch. A magnet is also useful for retrieving the ball bearings.
Preparation involves measuring the mass and radius of each ball bearing using digital scales and Vernier calipers, respectively, to calculate their density. The measuring cylinder is then filled with the test liquid and clamped securely to prevent tipping.
Rubber bands are strategically placed around the cylinder: the highest band must be far enough below the liquid surface to ensure the ball reaches terminal velocity before passing it. Two subsequent bands are placed 10-15 cm apart to allow for accurate timing of the terminal velocity segment.
The procedure involves carefully releasing a ball bearing into the liquid and starting a stopwatch when it passes the first rubber band. Lap timing or two separate stopwatches are used to record the time taken to fall between the marked distances. This process is repeated multiple times for each ball bearing diameter to ensure reliability and reduce random errors.
After each trial, the ball bearings are retrieved from the bottom of the cylinder using a magnet held against the outside wall. This allows for reuse and prevents contamination of the fluid or damage to the cylinder.

4. Data Analysis & Calculation

5. Key Distinctions: Laminar vs. Turbulent Flow

6. Exam Strategy & Tips

7. Error Analysis & Safety