A glass soda bottle is emptied of soda and filled to the very top with water. A

Question

A glass soda bottle is emptied of soda and filled to the very top with water. A cork is carefully fitted into the top of the bottle, leaving no air between the cork and the water. The top of the bottle has a diameter of D_top = 2.00 cm and the bottom of the bottle has a diameter of D_bot = 6.50 cm. The glass breaks when it is exposed to p_max = 70.0 MPa of pressure.
A student hits the cork sharply with her fist and the bottom of the bottle breaks. The student's fist has a mass of m = 0.480 kg and moves downward at a speed of v_i = 5.00 m/s. It collides elastically with the cork and rebounds with the same speed. The collision lasts for t = 1.20×10-4 s. In this problem, the positive direction is upward.

What is the force that the fist exerts on the top of the bottle?

Explanation / Answer

Given Rt=0.0100m, Rb=0.0325m, m=0.480kg, vi=-vf=5.00m/s, tc=120µs Find top and bottom areas: At = p(Rt)² = 0.0001p m² Ab = p(Rb)² = 0.00105625p m² Although water is incompressible, cork & fist are not. Assume deceleration is uniform and half the time of collision. First find force at top: Ft = m*dv/dt = m(vi-0)/(tc/2) = (0.48)(5)/((120*10^-6)/2) = 40 kN (or 9000 lbf) Now find force at bottom: Fb = Ft*Ab/At = (40000)(0.00105625p)/(0.0001p) = 422.5 kN Convert force to pressure to see if exceeds bottle rating: Pb = Fb/Ab = 127.3 MPa (or 18000 psi) 127.3 MPa > p_max= 70 MPa

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