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)
V=4/3 PI r^3
dV/dr=4PI r^2
dV=4PI r^2 dr
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?
F = G M1 M2/r^2
dF/dr = G M1 M2 [ -2 r /r^4 }
= - G M1 M2 (2/r^3)
Go back and understand the ones Bob and I have done.
(a) To express the relationship between the rate of change of volume of a sphere with respect to its radius, we can use the following differential equation:
dV/dt = k * r^2
Where:
- dV/dt represents the rate of change of volume with respect to time.
- k is the constant of proportionality.
- r is the radius of the sphere.
(b) To express the relationship between the change in gravitational attraction and the distance between two bodies, we can use the following differential equation:
dF/dx = -k' / x^3
Where:
- dF/dx represents the change in gravitational attraction with respect to the distance between the bodies.
- k' is the constant of proportionality.
- x is the distance between the bodies.
When calculating the gravitational attraction, we need the negative sign to indicate that the force is attractive.
For the constant of proportionality in each case:
(a) Since the rate of change of volume with respect to the radius being proportional to the square of the radius is a general property of spheres, we can use the constant of proportionality as k = 4/3π, where π is the mathematical constant pi.
(b) The constant of proportionality for gravitational attraction can vary depending on the units used for mass and distance. In many cases, it is common to use the value k' = G * m1 * m2, where G is the gravitational constant and m1, m2 are the masses of the bodies.