Question 180: Properties of Matter

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 :

Accepted Answer

Bulk modulus, B  =  $${{\Delta P} \over {{{\Delta V} \over V}}}$$
V
i
=  $$\pi $$ (a + $$\Delta $$a)
2
b
V
f
=  $$\pi $$a
2
b
$$ 	herefore $$   $$\Delta $$V = 2$$\pi $$ab$$\Delta $$a
$$ 	herefore $$   $${{{\Delta V} \over V}}$$ = $${{2\pi ab\Delta a} \over {\pi {a^2}b}}$$ = $${{2\Delta a} \over a}$$
$$ 	herefore $$   $$\Delta $$P = B $$ 	imes $$ $${{{\Delta V} \over V}}$$
= B $$ 	imes $$ $${{2\Delta a} \over a}$$
Normal Force (N) = $$\Delta $$P $$ 	imes $$ A
= B $$ 	imes $$ $${{2\Delta a} \over a}$$ $$ 	imes $$ (2$$\pi $$a)b
=   4$$\pi $$B$$\Delta $$ab
$$ 	herefore $$   Frictional force = $$\mu $$N =  (4$$\pi $$$$\mu $$Bb)$$\Delta $$a

Answer

($$\pi $$ $$\mu $$ B b) $$\Delta $$a

Answer

(2$$\pi $$ $$\mu $$ B b) $$\Delta $$a

Answer

($$\pi $$ $$\mu $$ B b) a

Answer

(4$$\pi $$ $$\mu $$ B b) $$\Delta $$a

Properties of Matter

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