Most Accurate Estimate of Surface Area of a Bottle?

Click For Summary
SUMMARY

The most accurate method to estimate the surface area of a cylindrical bottle that narrows at the neck involves practical techniques rather than theoretical calculations. Users can wrap the bottle in scotch tape to measure the length of tape used, which approximates the surface area. Alternatively, a mass balance can be utilized to measure the mass of a viscous fluid adhering to the bottle, allowing for calculations based on a known area. For bottles with a conical shape, formulas for the surface area of a frustum of a cone should be referenced for precise measurements.

PREREQUISITES
  • Understanding of basic geometry, specifically cylindrical and conical shapes.
  • Familiarity with practical measurement techniques, such as using scotch tape and mass balances.
  • Knowledge of surface area formulas, particularly for frustums of cones.
  • Experience with fluid dynamics concepts, including displacement and viscosity.
NEXT STEPS
  • Research the surface area formulas for frustums of cones.
  • Learn about practical measurement techniques for irregular shapes.
  • Explore the relationship between fluid displacement and surface area calculations.
  • Investigate the properties of viscous fluids and their applications in measurement.
USEFUL FOR

This discussion is beneficial for engineers, product designers, and anyone involved in physical measurements of irregular objects, particularly those working with bottles or similar shapes.

CAF
Messages
6
Reaction score
0
What is the most accurate way to estimate the surface area of a cylindrical bottle that decreases in diameter from its wides point (the body) to the narrowest point ( the neck and cap)?

It seems like we did similar problems in calculus, however: 1. I don't remember any of it, and 2. I think you were always provided with the mathmatical expression of curve of the bottle so that you could calculate it by rotating it around the center axis or something.

I just have the actual physical bottles in front of me and want to know what the outer surface areas are.

My best guess so far is to just treat them as cylinders and not take into account the curved surface where it transitions from the body to the neck, but I would like to be as accurate as possible.

Is there some correlation between displacement of water and surface area I could calculate by dunking them?

Any and all suggestions are appreciated.:smile:
 
Physics news on Phys.org
There is no fixed relationship between the volume and the surface area. This will depend on the actual shape of the bottle. You'll have to resort to some practical approach:

1. Find some scotch tape that comes in a tiny width. Carefully wrap the bottle in tape (avoiding gaps and overlaps) and keep track of the length of tape used.

2. If you've got a mass balance with a good sensitivity, you can do a dip in a viscous fluid (like honey) and measure the mass of fluid on the bottle. Then using the mass stuck to a calibrated known area of identical glass you can find the area of the bottle.

3. Variation of above technique : cover bottle in sticky goop and roll it in a pan of tiny ball bearings (or equivalent). The balls will make a monolayer thick coating.
 
It is not clear how you can have a "cylinder" that decreases in diameter. Cylinders have a constant diameter by definition. Perhaps you have a cone? In that case you can look up formulas for the surface area of a cone, Google "frustum of a cone"
 
Here is an example of the type of bottle I am referring to:
 

Attachments

  • ProductsMoldedBottles.jpg
    ProductsMoldedBottles.jpg
    7.4 KB · Views: 699

Similar threads

Undergrad Question about Integrals to Determine Volume vs Surface Area
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Gabriel's Horn, Inside vs. Outside Surface area
  • · Replies 9 ·
Replies
9
Views
3K
Undergrad The surface area of an oblate ellipsoid
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad Surface formed by moving area along curved axis
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Find the area enclosed by the curve y = x csch(x+y)
  • · Replies 3 ·
Replies
3
Views
2K
High School Use Surface-Volume to approximate gravity for planets and protons
  • · Replies 6 ·
Replies
6
Views
1K
Undergrad The Basic Area Problem (introduction to the topic of integrals)
  • · Replies 24 ·
Replies
24
Views
4K
High School Oriented Surfaces & Surface Area: Investigating the Impact
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Calc II: i don't understand integral for surface area
  • · Replies 5 ·
Replies
5
Views
2K
MHB How can the area of a conic surface be calculated in two different ways?
  • · Replies 2 ·
Replies
2
Views
1K