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Preliminary Mathematics
Published in P.K. Jayasree, K Balan, V Rani, Practical Civil Engineering, 2021
P.K. Jayasree, K Balan, V Rani
A sphere is a 3D object where all the points on the surface of the object are equidistant from a common center. If d is the diameter, r is the radius, and C is the circumference of the sphere, Surfacearea,A=πd2=4πr2=Cd=0.3183C2Volume,V=(A/6)d=πd3/6=0.5236d3=4/3πr3
A review of thermodynamic concepts
Published in Ronald L. Fournier, Basic Transport Phenomena in Biomedical Engineering, 2017
Recall that for a sphere of radius r, the surface area is 4πr2. For a soap bubble, there are two interfaces that have a change in area, the inside and outside surfaces of the liquid film, so dAS in this case is equal to d(2 × 4πr2) = 16πr dr. For a droplet or a gas bubble, where we have either a drop of liquid suspended in a gas, or a gas bubble suspended in a liquid, there is only one interface and dAS = d(4πr2) = 8πr dr. Substituting in for dAS and AS in Equation 2.183, we get Equations 2.184 and 2.185 for the pressure difference PB − PA: 2interfacesPB−PA=4γr1interfacePB−PA=2γrEquations 2.184 and 2.185 are known as the Laplace-Young equation and shows that the excess pressure (PB − PA) is inversely proportional to the radius of the bubble or the droplet. The radius r in the above equations is always considered to have its center of curvature in the phase in which PB is measured. Hence, for a gas bubble in a liquid, PB is the pressure within the bubble, which is then greater than the pressure (PA) in the liquid phase that surrounds the bubble.
Calculus of pasta, sausages, and bagels: can their surface areas be derivatives of their volumes?
Published in International Journal of Mathematical Education in Science and Technology, 2020
Let us now return to the torus. As we have pointed out, a torus is the result of sliding a sphere with radius r along a circle of radius R. However, a disk of the same radius, r, moving along this circle such that the plane of the disk is normal to the plane of the circle produces exactly the same torus. The volume of the torus is therefore the product of the disk area and the circumference of the ‘large’ circle , and the surface area of the torus is the product of the disk circumference and the circumference of the ‘large’ circle that is the length of the closed curve.