In an experiment to demonstrate the force due to air pressure, Otto von Guericke

Question

In an experiment to demonstrate the force due to air pressure, Otto von Guericke (1602- 1686) evacuated a sphere made of two thin-walled hollow hemispheres of radius R. The component along the z-axis of the force due to the pressure acting on the shaded strip in the figure is dFP,2 = -(P0 - P)dS cos(theta), where dS is the area of the strip, P0 the atmospheric pressure and P the residual pressure inside the sphere. Derive an expression for dS as a function of R, theta and d theta and hence show that dFP, z = -(P0 - P)2piR2sin(theta)cos(theta)d theta. Compute the total z-component of the force exerted on one hemisphere by the pressure difference, by integrateting the expression for the force obtained in (a) over theta from 0 to pi/2. Determine the force needed to pull the two hemispheres apart, for P=0.1 P0 and R=0.3 m.

Explanation / Answer

a)

dS = (2R sin) * (R d)

     = 2R2 sin d

dF = -(P0 - P) dS cos d = -(P0 - P) 2R2 sin cos d

b)

Integration from 0 to /2

F = -(P0 - P) 2R2 sin cos d

   = -(P0 - P) 2R2 sin cos d

    = -(P0 - P) 2R2 ((cos)2/2)

    = -(P0 - P) R2 (cos)2

    = -(P0 - P) R2 (cos(/2))2 - (-(P0 - P) R2 (cos0)2)

    = (P0 - P) R2 (cos0)2

     = (P0 - P) R2

c)

P0 = 1.013e5 Pa

F = (P0 - P) R2 = (1.013e5 - 0.1*1.013e5)*(3.14*0.3*0.3) = 2.58 * 10^4 N

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