an object is located a distance d above the surface of a large planet of radius r. at this position, its true weight is one percent less than its true weight on the surface. what is the ratio of d/r?
To find the ratio of d/r, we can start by understanding the concept of the true weight of an object at different distances from the surface of a planet.
The true weight of an object is the force of gravity acting on it. This force depends on the mass of the object and the gravitational acceleration of the planet.
On the surface of a planet, the gravitational acceleration is given by g = GM/r^2, where G is the gravitational constant, M is the mass of the planet, and r is its radius.
When an object is located a distance d above the surface, the gravitational acceleration at that point is slightly lower than on the surface. Let's denote this reduced gravitational acceleration as g'.
The difference in gravitational acceleration between the surface and a distance d above it can be calculated using the inverse square law. The inverse square law states that the gravitational acceleration decreases with the square of the distance.
Therefore, we have the following relationship:
g' = g(1 - (d/r)^2)
Given that the true weight of the object at a distance d above the surface is one percent less than its true weight on the surface, we can set up the following equation:
g' = 0.99g
Substituting the expressions for g' and g, we get:
0.99g = g(1 - (d/r)^2)
Simplifying the equation, we have:
0.99 = 1 - (d/r)^2
Rearranging the equation, we get:
(d/r)^2 = 1 - 0.99
(d/r)^2 = 0.01
Taking the square root of both sides, we find:
d/r = 0.1
Therefore, the ratio of d to r is 0.1, or 1:10.