Question

4. Write down a differential equation to express each of the following relationships:
(a) The rate at which the volume of a sphere changes with radius is proportional to the square of the radius.
(b) Gravitational attraction F between two given bodies of masses m1 and m2 decreases with the distance between them. The change in gravitational attraction is proportional to the inverse of the cube of the distance between them.

Suggest a value for the constant of proportionality in each case(for part(b),you might like to consult)

Answer 1

V=4/3 PI r^3

dV/dr=4PI r^2
dV=4PI r^2 dr

Answer 2

V = (4/3) pi r^3
dV/dr = (4/3) 3 pi r^2

dV/dr = 4 pi r^2

OF COURSE this is the surface area of the sphere, see why?

Answer 3

F = G M1 M2/r^2

dF/dr = G M1 M2 [ -2 r /r^4 }

= - G M1 M2 (2/r^3)

Answer 4

Go back and understand the ones Bob and I have done.