Question

A bottle has an opening of radius $a$ and length $b$. A cork of length $b$ and radius $(a+ \Delta a)$ where $(\Delta a < < a)$ is compressed to fit into the opening completely (See figure). If the bulk modulus of cork is $B$ and frictional coefficient between the bottle and cork is $\mu$ then the force needed to push the cork into the bottle is :

• A. $(2 \pi \mu B\, b) \Delta\,a$
• B. $(\pi \mu B\, b) \Delta a$
• C. $(\pi \mu \,B\, b ) a$
• D. $(4 \pi \mu\, B \,b ) \Delta \,a$

Answer 1

The Correct Option is D

 The problem involves understanding the mechanics of fitting a cork into a bottle where the cork has a slightly larger radius than the bottle opening. Here's a step-by-step explanation to determine the force required:

1. First, understand that the cork, initially having a radius of \(a + \Delta a\), is compressed to fit into the bottle opening of radius \(a\).
2. The compression of the cork involves a change in the radial dimension, which can be expressed using the formula for volume change relation with bulk modulus. The change in volume \(\Delta V\) for a cylinder cork is primarily due to the change in radius. The cork is compressed to a radius \(a\) from \(a + \Delta a\).
3. The bulk modulus \(B\) is defined as: \(B = -\frac{\Delta P}{\Delta V/V}\), where \(\Delta P\) is the change in pressure and \(\Delta V\) is the change in volume.
4. The force required to compress the cork against this bulk modulus is directly related to the surface contact area of the cork and the pressure exerted by this compression. The pressure exerted by the cork can be driven by: \(F = \text{frictional force} = \mu \times \text{normal force}\)
5. Considering the frictional force holds the cork in position, the contact force relates to the entire lateral surface area of the cork compressed, which is:
6. Given the friction coefficient \(\mu\), the actual force necessary to overcome the friction and compress the cork entirely is: .