Question

The pressure inside a bottle of champagne is 4.5 atm higher than the air pressure outside. The neck of the bottle has an inner radius of \(1.0 \mathrm{cm} .\) What is the frictional force on the cork due to the neck of the bottle?

Short answer

The frictional force on the cork is approximately 143.2 N.

Step-by-step solution

Understanding the Problem

We need to find the frictional force on the cork due to the pressure difference inside and outside the bottle. The pressure inside the bottle is 4.5 atm higher than the outside. The data provided includes the inner radius of the bottle's neck, which is 1.0 cm.

Converting Units

First, convert the pressure from atmospheres to pascals (SI units of pressure). Use the conversion factor: 1 atm = 101325 Pa. Therefore, the pressure difference is \(4.5 \times 101325 = 456000\; \text{Pa}\).

Calculating the Area of the Bottle’s Neck

The area of a circle is given by the formula \( A = \pi r^2 \). Here, \( r = 1.0\; \text{cm} = 0.01 \; \text{m} \). Substitute to find the area: \( A = \pi \times (0.01)^2 = 3.14 \times 10^{-4} \; \text{m}^2 \).

Finding the Force Due to Pressure

Force is pressure times area (\( F = PA \)). Substitute the values: \( F = 456000 \; \text{Pa} \times 3.14 \times 10^{-4} \; \text{m}^2 = 143.184 \; \text{N} \). This is the normal force exerted by the bottle's neck on the cork due to the pressure difference.

Calculating Frictional Force (No Movement)

Assuming no movement of the cork, the frictional force balances the normal force. If static friction is considered, it equals the force calculated. Hence the frictional force is 143.184 N since the cork stays in place without sliding out.

Key concepts

Pressure Difference

When discussing pressure difference, we refer to the difference in pressure between two separate points. In this context, it is the disparity between the pressure inside a champagne bottle and the external atmospheric pressure. This difference directly impacts how much force is being exerted on the cork.
\[\text{Pressure Difference} = \text{Inside Pressure} - \text{Outside Pressure}\]
In our exercise, the scenario involves an inside pressure which is 4.5 atm higher than the pressure outside. To make practical calculations, it is essential to convert this from atmospheres (atm) to pascals (Pa), since pascals are the SI unit of pressure.
This step is crucial because standard atmospheric pressure in pascals is known, making calculations more reliable. After conversion, understanding this concept aids us in determining the force applied to the cork as a result of this pressure disparity.

Circular Area Calculation

Calculating the area of a circular section, such as the neck of a bottle, often requires using the formula for the area of a circle.
\[A = \pi r^2\]
Here, the radius \(r\) needs to be in meters for any further calculations involving SI units. In our problem, the radius provided is 1.0 cm. To convert this to meters, we divide by 100, giving us 0.01 meters.
Once the radius is in the correct unit, we substitute it back into the formula to find the area. This calculation reveals how much surface area is under the influence of the pressure from inside the bottle.

• Radius of the neck: 0.01 m
• Area \(A\): \( \pi \times (0.01)^2 \)
• Result: \(3.14 \times 10^{-4} \; \text{m}^2\)

The circular area calculation is a vital step as it directly affects the force calculation by providing the effective area subjected to pressure.

Unit Conversion in Physics

Converting units is a crucial skill in physics, allowing us to translate quantities into universally understood dimensions. In many exercises, including ours, unit conversion ensures all calculations align to the scientific standard, usually SI units.
In this exercise:
- Pressure is originally in atmospheres (atm), a common unit outside the scientific community for measuring atmospheric pressure's baseline.
- We convert it to pascals (Pa), the standard SI unit.
Using the conversion factor \(1 \text{ atm} = 101325 \; \text{Pa}\), we find the pressure difference in pascals:

• Pressure difference: \(4.5 \text{ atm}\)
• In pascals: \(4.5 \times 101325 = 456000 \; \text{Pa}\)

Getting these conversions right not only solidifies understanding but ensures accuracy in equations, especially those involving derived units like force, which depend on consistent base units.

Normal Force

Normal force is crucial to understanding how forces interact on surfaces. It’s the perpendicular force exerted by a surface to support the weight of an object resting on it. However, in this problem, it refers to the pressure force acting perpendicularly at the neck of the bottle against the cork.
Normal force (\(F=N\)) due to pressure can be calculated as:
\[F = P \times A\]
where:- \(P\) is the pressure difference,- \(A\) is the area calculated previously.
The frictional force keeping the cork in place equals this calculated normal force due to pressure difference, given that the cork isn't moving. This balance is crucial in scenarios where static friction prevents sliding, revealing that the cork's stability entirely depends on this force.
Applying these principles, we calculate in the given scenario:

• \(F = 456000 \; \text{Pa} \times 3.14 \times 10^{-4} \; \text{m}^2 = 143.184 \; \text{N}\)

Understanding normal force helps you visualize how pressure differences can lead to real, measureable forces on objects.