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Step-by-Step Solution
Step 1: Understand the Situation
A cork of length $b$ and radius $(a + \Delta a)$ is compressed so that it fits into a bottle neck of radius $a$. The cork experiences a change in volume, and this compression leads to an increase in the pressure inside the cork. The force required to push the cork is due to the friction between the cork and the bottle neck, which depends on this increased pressure (normal force) and the coefficient of friction $\mu$.
Step 2: Write Down the Relevant Bulk Modulus Relation
The bulk modulus $B$ is defined by:
$ B = \frac{\Delta P}{\frac{\Delta V}{V}} $,
where $\Delta P$ is the change in pressure, $V$ is the original volume of the cork, and $\Delta V$ is the change in volume.
Step 3: Identify the Initial and Final Volumes of the Cork
Initial volume of the cork, $V_{i}$:
$ V_{i} = \pi \,(a + \Delta a)^{2}\, b $
Final volume of the cork, $V_{f}$:
$ V_{f} = \pi \,a^{2}\, b $
Step 4: Calculate the Change in Volume
The change in volume $\Delta V$ is:
$ \Delta V = V_{i} - V_{f} = \pi \bigl((a + \Delta a)^{2} - a^{2}\bigr)\,b $
Expanding and keeping terms with $\Delta a$ (assuming $\Delta a \ll a$) gives approximately:
$ \Delta V \approx \pi \bigl(a^{2} + 2a\,\Delta a - a^{2}\bigr)\,b = 2 \pi a\, b\, \Delta a
Step 5: Compute the Fractional Change in Volume
The fractional change in volume is:
$ \frac{\Delta V}{V} = \frac{2 \pi a\, b\, \Delta a}{\pi a^{2}\, b} = \frac{2\, \Delta a}{a}.
Step 6: Find the Change in Pressure
From the definition of bulk modulus:
$ \Delta P = B \times \frac{\Delta V}{V} = B \times \frac{2\, \Delta a}{a}.
Step 7: Calculate the Normal Force
The normal force $N$ on the cork is:
$ N = \Delta P \times \text{(Contact Area)}.
Here, the contact area between the cork and the bottle neck is the lateral surface area during compression, which is approximately $2 \pi a \times b$ (circumference $2\pi a$ times length $b$):
$ N = \left( B \times \frac{2\,\Delta a}{a} \right) \times \left(2 \pi a \, b\right).
Simplify:
$ N = 4 \pi B \, \Delta a \, b.
Step 8: Determine the Force Needed to Push the Cork
The frictional force $F$ (the force required to push the cork) is given by:
$ F = \mu \, N = \mu \times \bigl(4 \pi B\, \Delta a\, b\bigr).
Therefore,
$ \boxed{ F = 4 \pi \mu \, B \, b \,\Delta a. }
Final Answer
$\displaystyle (4 \pi \, \mu \, B \, b)\,\Delta a$
