Core Practical 4: Investigating Viscosity

Force Balance: At terminal velocity, the net force on the sphere is zero. This is expressed as $W = U + F$, where $W$ is weight, $U$ is upthrust (buoyancy), and $F$ is the viscous drag force.

Mathematical Derivation: Substituting the formulas for weight ($mg$), upthrust (weight of displaced fluid), and Stokes' drag ($6\pi\eta rv$), we get: $\frac{4}{3}\pi r^3 \rho_s g = \frac{4}{3}\pi r^3 \rho_f g + 6\pi\eta r v_t$

Viscosity Formula: Rearranging the force balance for viscosity $\eta$ yields: $\eta = \frac{2r^2g(\rho_s - \rho_f)}{9v_t}$ where $r$ is the radius, $g$ is gravity, $\rho_s$ is sphere density, and $\rho_f$ is fluid density.

Proportionality: The derivation shows that for a specific fluid and material, the terminal velocity $v_t$ is directly proportional to the square of the radius ($r^2$).

Measuring Physical Constants: Use a micrometer screw gauge to measure the diameter of the spheres (calculating $r$) and a balance to find the mass of the spheres and a known volume of the liquid to determine densities $\rho_s$ and $\rho_f$.

Establishing Terminal Velocity: Mark the cylinder with rubber bands at regular intervals. Ensure the first mark is far enough below the surface so the sphere has already reached its constant terminal velocity before timing begins.

Timing and Recording: Drop spheres of varying radii and record the time taken to pass between the markers. Repeat measurements for each size to calculate a mean time and reduce random error.

Graphical Analysis: Plot a graph of $v_t$ (y-axis) against $r^2$ (x-axis). The gradient of the resulting straight line through the origin is equal to $\frac{2g(\rho_s - \rho_f)}{9\eta}$, allowing for a precise calculation of $\eta$.

Weight vs. Upthrust: Weight depends solely on the mass of the falling object, whereas upthrust depends on the volume of the object and the density of the fluid it displaces.

Instantaneous vs. Terminal Velocity: Instantaneous velocity increases as the object accelerates from rest; terminal velocity is the specific constant value reached when acceleration becomes zero.

Verify Terminal Velocity: Always explain how you ensured terminal velocity was reached, such as by checking if the time taken to travel between consecutive equal distances remains constant.

Wall Effect Corrections: In exams, be aware that if the cylinder is narrow, the 'wall effect' slows the sphere down. A wider cylinder is preferred to approximate an 'infinite' fluid as required by Stokes' Law.

Units Consistency: Ensure all measurements are converted to SI units (meters, kilograms, seconds) before calculation. Viscosity is measured in Pascal-seconds ($Pa \cdot s$) or $kg \cdot m^{-1} \cdot s^{-1}$.

Temperature Control: Viscosity is highly temperature-dependent. Mention that the temperature of the liquid should be kept constant or recorded, as an increase in temperature typically decreases the viscosity of a liquid.

Density Confusion: Students often forget to subtract the fluid density from the sphere density. The effective driving force is the 'excess' weight ($W - U$), not just the weight itself.

Parallax Error: When timing the sphere as it passes markers, the observer's eye must be level with the marker to avoid timing errors that lead to inaccurate velocity calculations.

Small Sphere Limitation: Stokes' Law only applies if the sphere is small enough that the flow remains laminar. If the sphere falls too fast, turbulence occurs and the linear relationship $v \propto r^2$ breaks down.