Two objects of equal mass m are separated by a distance r. Which expression gives the gravitational potential at the point P mid way between the two objects?
The answer shows as -4Gm/r, but I can't see where this comes from.
Midway between the two objects, you add the gravitational potentials due to each object. The distance to each object is r/2.
The PE due to each separate mass is
-Gm/(r/2) = -2Gm/r
Double that for two objects at the same distance, and you have the "book" answer.
Note that "gravitational potential" is gravitational potential energy per unit mass placed at that location. The units are joules per kg
To find the gravitational potential at the point P midway between two objects of equal mass, you can use the principle of superposition.
The gravitational potential at a point due to an object is given by the formula:
V = - Gm/r
Where:
V is the gravitational potential
G is the universal gravitational constant (approximately 6.67430 × 10^-11 N m^2/kg^2)
m is the mass of the object
r is the distance between the object and the point
Now, considering the two objects of equal mass (m) separated by a distance (r), the gravitational potential at point P can be calculated by summing the gravitational potentials due to each object individually.
The object on the left is at a distance r/2 from point P, and the object on the right is also at a distance r/2 from point P. Using the formula above, the gravitational potential due to each object can be written as:
V_left = - Gm/(r/2)
V_right = - Gm/(r/2)
The total gravitational potential at point P is given by the sum of these two potentials:
V_total = V_left + V_right
= - Gm/(r/2) - Gm/(r/2)
= - 2Gm/(r/2)
= - 4Gm/r
Therefore, the expression for the gravitational potential at point P midway between the two objects is -4Gm/r.