Introduction: The Science of "Thick" Liquids

Have you ever noticed how honey pours much more slowly than water? Or how a marble drops quickly through a glass of water but crawls through a jar of syrup? This "thickness" or resistance to flow is a physical property called viscosity. In this core practical, we use a falling ball to measure exactly how viscous a liquid is.

Don't worry if the formulas look a bit intimidating at first! We are essentially just balancing three forces to see how much the liquid "fights back" against a moving object. Understanding this is vital for everything from engine oil performance to how blood flows through our veins.

The Physics Behind the Experiment

When a small sphere falls through a liquid, it eventually reaches a constant speed called terminal velocity \( (v) \). At this point, the forces acting downwards are perfectly balanced by the forces acting upwards.

1. The Three Forces

Weight (\( W \)): Acts downwards. Calculated as \( W = mg \). Since mass is density \( \times \) volume, and the ball is a sphere: \( W = \rho_{s} \frac{4}{3} \pi r^3 g \), where \( \rho_{s} \) is the density of the sphere.

Upthrust (\( U \)): Acts upwards. According to Archimedes' Principle, this is the weight of the fluid displaced: \( U = \rho_{l} \frac{4}{3} \pi r^3 g \), where \( \rho_{l} \) is the density of the liquid.

Drag Force (\( F \)): Acts upwards. For a sphere moving slowly through a liquid, we use Stokes' Law: \( F = 6 \pi \eta r v \).

2. Stokes' Law Conditions

It is very important to remember that Stokes' Law only applies if:

  • The object is a small sphere.
  • It is moving at a low speed.
  • The flow is laminar (smooth layers, no bubbles or swirls).

3. The Equation for Viscosity

At terminal velocity: Forces Up = Forces Down
\( U + F = W \)
\( \rho_{l} \frac{4}{3} \pi r^3 g + 6 \pi \eta r v = \rho_{s} \frac{4}{3} \pi r^3 g \)

By rearranging this, we can solve for the viscosity coefficient \( (\eta) \). Usually, we use a graph to make this more accurate.

Quick Takeaway: Viscosity \( (\eta) \) is a measure of internal friction in a fluid. Higher viscosity means the liquid is "thicker" and resists flow more.

Apparatus and Measurements

To get a high-quality result, you need to use the right tools for the right job. This is a major focus for your Unit 3 exam!

  • A tall wide measuring cylinder: Filled with the liquid (e.g., heavy oil or glycerol). It needs to be tall so the ball has time to reach terminal velocity.
  • Ball bearings: Of various small diameters.
  • Micrometer Screw Gauge: Used to measure the diameter of the ball bearings. It has a resolution of 0.01 mm.
  • Stopwatch: To time the fall between two markers.
  • Metre ruler: To measure the distance between markers.
  • Digital Balance: To find the mass of the balls and the liquid (to calculate density).

Step-by-Step Procedure

Step 1: Determine Densities
Measure the mass of an empty measuring cylinder, then add a known volume of liquid to find its mass. Use \( \rho = m/V \) to find the liquid's density. For the ball bearings, measure their mass and use the micrometer for the radius to find their density.

Step 2: Set Up Markers
Place rubber bands around the cylinder. The first marker must be far enough down so the ball has already reached terminal velocity before you start the timer.

Step 3: Drop and Time
Drop a ball bearing into the center of the cylinder. Start the timer when the bottom of the ball passes the first marker and stop it when it passes the second marker.

Step 4: Repeat
Repeat this for different diameters of ball bearings. Always do repeat readings for each ball to improve precision and identify anomalies.

Did you know? Temperature affects viscosity significantly. As a liquid gets hotter, it usually becomes less viscous (thinner). Keep the room temperature constant during your experiment!

Data Processing and Analysis

The most common way to find viscosity from your data is by plotting a graph. From the force balance equation, terminal velocity \( (v) \) is proportional to the square of the radius \( (r^2) \).

The Graph: Plot terminal velocity \( v \) on the y-axis against \( r^2 \) on the x-axis.
The Gradient: The gradient of this straight line is related to the viscosity by the formula:
\( \text{gradient} = \frac{2g(\rho_{s} - \rho_{l})}{9\eta} \)

To find \( \eta \), you simply rearrange: \( \eta = \frac{2g(\rho_{s} - \rho_{l})}{9 \times \text{gradient}} \)

Key Tip for Exams: When calculating the gradient, always use a large triangle on your graph (covering at least half the line) to reduce uncertainty.

Uncertainties and Errors

In Physics exams, you are often asked how to make the experiment better. Here are the common "traps" and fixes:

  • Zero Error: Always check if your micrometer or digital balance reads \( 0.00 \) before you start. This is a systematic error.
  • Parallax Error: Ensure your eyes are level with the markers when starting/stopping the stopwatch to avoid timing errors.
  • Edge Effects: If the ball is too close to the wall of the cylinder, the flow isn't truly "infinite," and extra drag is created. Use a wide cylinder and drop the ball down the center.
  • Timing Uncertainty: Human reaction time is a factor. Using a longer distance between markers reduces the percentage uncertainty in the time measurement.

Safety First!

While dropping balls into oil seems safe, remember:

  • Spilled oil: Is an extreme slip hazard. Clean up any drops immediately.
  • Heavy cylinders: Are easy to knock over. Ensure they are stable on the bench.

Quick Review

Stokes' Law: \( F = 6 \pi \eta r v \) (Only for small spheres, slow speeds, laminar flow).
At Terminal Velocity: Weight = Upthrust + Drag.
Viscosity Change: Higher temperature usually means lower viscosity for liquids.
Precision: Use a micrometer (0.01 mm resolution) and repeat readings to reduce random error.