# Physics Lab Experiment- Confused on what we are measuring

• taculator
In summary, in a physics class assessment, students designed a lab involving plastic cups with different diameter holes and measured the flow time of various liquids. They questioned if they were measuring flow rate or another value and how to construct a lab write up based on the density and diameter variables. The discussion then shifted to the importance of viscosity in fluid dynamics and the dimensions and units of viscosity. The problem was deemed complex, but could be analyzed using dimensional analysis and the scaling laws in hydrodynamics.
taculator
In my physics class for an assessment we designed our own lab.

In our lab we had plastic cups and cut small holes in the bottom ranging from 0.3 cm to 1.3 cm diameters. We then measured the amount of time it took for different volumes of water, detergent, syrup and rubbing alcohol. I understand that each of these liquids have a different density and that the amount of time for each substance to flow out of the cup will vary.

Here is my question:
In this lab, are we measuring flow rate or another value? How exactly can i construct a lab write up based on these variables:

the density of the liquids vs the time it took to flow out
the diameter of the cups vs the time it took to flow out
the density of the liquids vs the diameter of the cups?

When a fluid flows through a constriction, another important characteristic of the fluid is the so called viscosity. It enters in Newton's Law of viscous force. Namely, if you have two parallel sheets of the fluid a distance $\Delta l$ apart, flowing with a difference in velocities $\Delta v$, then the tangential force felt by each one is proportional to the common area $A$ of the two sheets, the above mentioned velocity gradient $\Delta v/\Delta l$. The coefficient of proportionality is the dynamical viscosity $\eta$ of the fluid:
$$F = \eta \, A \, \frac{\Delta v}{\Delta l}$$
Of course the direction of this force is opposite on each sheet. It tends to slow down the faster moving sheet, and accelerate the more slowly moving sheet.

From the above law, you can find the dimensions of viscosity:
$$\left[ \eta \right] = \frac{[ F ] \, [ l ]}{ [ A ] \, [ v ] } = \frac{\mathrm{T}^{-2} \, \mathrm{L} \, \mathrm{M} \, \mathrm{L} } { \mathrm{L}^2 \, \mathrm{T}^{-1} \mathrm{L} } = \mathrm{T}^{-1} \, \mathrm{L}^{-1} \, \mathrm{M}$$

Because $F/A$ has the meaning of a strain, with the same dimension as pressure, and the velocity gradient has the dimensions of $[v]/[l] = \mathrm{T}^{-1} \mathrm{L}/\mathrm{L} = \mathrm{T}^{-1}$ inverse time, the dynamical viscosity is usually reported in units of $[\mathrm{pressure}] \cdot [\mathrm{time} ]$.

Now, your problem is a pretty complicated hydrodynamics problem. Nevertheless, you may draw a series of conclusions by employing arguments from dimensional analysis. The "fundamental constants" that govern this process are:
1) acceleration due to Earth's gravity $g, \ [g] = \mathrm{T}^{-2} \mathrm{L}$

2) density of the liquid, $\rho, \ [\rho] = \mathrm{L}^{-3} \mathrm{M}$

3) viscosity of the liquid, $\eta, \ [\eta] = \mathrm{T}^{-1} \mathrm{L}^{-1} \mathrm{M}$
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Then, there is the parameter of each cup, namely the diameter of its hole $D, \ [D] = \mathrm{L}$

A quantity which you experimentally measure is the flow time $T, [ T ] = \mathrm{T}$

By dimensional analysis, you can construct a combination of the first three constants with the dimensions of length $L_0$ and time $T_0$, respectively. Find these!

Then, you can form the dimensionless variables $D/L_0$, and $T/T_0$. It is a consequence of the scaling laws in hydrodynamics that there is a general functional relationship:

$$\frac{T}{T_0} = f \left( \frac{D}{L_0} \right)$$

You can use it, in principle to measure the viscosity of an unknown fluid in terms of the viscosity of a known fluid, say water. I will let you figure out what viscosity may be measured in this way.

## 1. What is the purpose of this physics lab experiment?

The purpose of this experiment is to measure and analyze physical phenomena in order to better understand the fundamental laws and principles of physics.

## 2. What are we supposed to be measuring in this experiment?

In this experiment, we will be measuring various physical quantities such as mass, length, time, and temperature. These measurements will be used to calculate and analyze other physical properties.

## 3. How do we know if our measurements are accurate?

To ensure accuracy, it is important to follow proper experimental procedures and use calibrated instruments. Additionally, repeating the experiment multiple times and comparing results can help verify the accuracy of our measurements.

## 4. What should we do if we are confused about the experimental setup or procedure?

If you are confused about any aspect of the experiment, it is important to ask your instructor or lab partner for clarification. It is also helpful to review the lab manual and any accompanying resources before beginning the experiment.

## 5. How can we use the data we collect in this experiment?

The data collected in this experiment can be used to analyze and understand the relationship between different physical quantities, as well as to test and verify various theories and laws in physics. It can also be used to make predictions and draw conclusions about real-world phenomena.

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