A certain planet is a uniform sphere of mass M and radius R of 5.1 x 10^6 m. A Mountain on the surface of the planet has a height of 2000 m. Suggest why the value of the gravitational field strength at the base of the mountain and at the top of the mountain are almost equal. Use words and equations
The value of the gravitational field strength depends on the distance from the center of mass of an object. However, when measuring the gravitational field strength on the surface of a planet, the heights involved are negligible compared to the radius of the planet.
The equation for the gravitational field strength (g) is given by:
g = (G * M) / R^2
Where:
- G is the gravitational constant
- M is the mass of the planet
- R is the radius of the planet
In this case, the difference in height for the base and top of the mountain (2000 m) is much smaller compared to the radius of the planet (5.1 x 10^6 m). Therefore, the difference in distance from the center of mass is negligible, and the gravitational field strength at the base and top of the mountain is almost equal.
Mathematically, the difference in the gravitational field strength between the base and top of the mountain can be calculated as:
Δg = g_top - g_base
Using the equation for g mentioned earlier, we can write:
Δg = [(G * M) / R^2_top] - [(G * M) / R^2_base]
Since R^2_top and R^2_base are nearly equal (due to the negligible difference in height), we can say that:
Δg ≈ 0
Therefore, the value of the gravitational field strength at the base and top of the mountain are almost equal.