answer is 288.7 micrometers and -126.7 Pa please help with steps Two glass micro

Question

answer is 288.7 micrometers and -126.7 Pa please help with steps

Two glass microscope slides are separated by a spacer in such a way that the slides are parallel to each other, with a separation of 500 mum. This glass "sandwich" is placed in a vertical orientation, with one open end slightly below the surface of a reservoir of silicone oil. Take sigma = 0.038 N/m and p = 900 kg/m3. Assume that silicone oil has a contact angle of 30 degree on glass. If we approximate the meniscus as a part of a cylindrical surface, the radius of curvature of the meniscus is If the radius of curvature of the meniscus is 300 mum, the gage pressure at the meniscus in the oil phase is

Explanation / Answer

a) If u just draw the diagram it is clear that for cylindrical miniscus radius of curvature r=t/(2*cos30)

so we get r = 288.7 micrometers (here t = 500 micrometers)

b) here since the miniscus is conave upwards we know pressure is more above so gage pressure below the miniscus is negative 

and this difference is due to the surface tension force which is acting upwards and it is 2Tlcos30

here T is surface tension l is length of the plate perpendicular to the force (here the horizontal component gets nullified)

and this should be equal to the difference in pressure forces acting above and below the miniscus 

for the miniscus to be in equilibrium

and the difference in pressures is gagepressure * effective area of crosssection 

To make it easier lets consider the equilibrium of the miniscus

we get 

pressure force from above the miniscus(lets call it Fpa)-surface tension force(S) = pressure force below the miniscus(Fpb)(negative sign since surface tension force is oppisite) ------------>(1)

surface tension force = 2Tlcos30

first we calculate pressure force from above then we can calculate pressure force from below in the same way

calculating pressure force from above

consider miniscus and and consider a strip  of agular width dtheta and making an angle theta with vertical

and of length l

two such strips can be selected one towards right and other towards left

u can see the pressure force acts perpendicuar to these strips and their horizontal components gets cancelled

and u integrate this with respect to theta to get net pressure force from above from the limits (limits for angle can be found geometrically they are from 0 to 60 degrees)

u get Fpa = 2prlcos30 = ptl so we can find this way or else i have said it is pressure * effective area of crossection and here it is projection of that curved surface area onto a plane surface that is tl

but effective area of crosssection concept is used when pressure is uniform so first is better method

here p is pa that is atmosperic pressure

so Fpb = 2pab*tl(pab is absolute pressure) = 2(p+pg)*tl (pg is gage pressure)

if u substitute in eq (1) we get pa = -T/r = -126.7Pa

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