A planet is growing by accretion of material from the solar nebula. Suppose that as it grows, its density remains constant. Does the force of gravity at its surface increase, decrease, or stay the same? Why?
Also, What would happen to the surface gravity as the radius of the planet doubled? Why?
The mass M is then proportional to the volume, so it is proportional to the third power of the radius. The force of gravity F at the surface is inversely proportional to the square of the radius and proportional to the mass:
F = proportional to M/R^2 = R
cannot understand your point
Are you saying that as the radius of the planet doubled the surface gravity would double too and it is because of the answer you gave before
The static value of gravity on, or above the surface of a spherical body is directly proportional to the mass of the body and inversely proportional to the square of the distance from the center of the body and is defined by the expression g = GM/r^2 = µ/r^2 where GM = µ = the gravitational constant of the body (G = the Universal Gravitational Constant and M = the mass of the body) and r = the distance from the center of the body to the point in question.
This should enable you to reach the conclusions you seek.
To determine how the force of gravity changes as a planet grows, we need to consider Newton's law of universal gravitation, which states that the force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
In this scenario, as the planet grows by accreting material from the solar nebula, its mass increases while its radius also increases. Given that the density remains constant, we can assume that the mass is proportional to the cube of the radius (since the volume is proportional to the cube of the radius when the density is constant). So, when the radius doubles, the mass of the planet increases by a factor of 8 (2^3).
With this information in mind, let's examine the two parts of your question:
1. Does the force of gravity at the planet's surface increase, decrease, or stay the same as it grows?
To determine the force of gravity at the planet's surface, we need to calculate the gravitational acceleration (g). The formula for gravitational acceleration is g = G * (M / r^2), where G is the gravitational constant, M is the mass of the planet, and r is the radius of the planet.
Since the mass of the planet is increasing as it grows, while the radius is also increasing, we can conclude that the force of gravity at the planet's surface will increase. The increase in mass has a greater effect on the force of gravity than the increase in radius.
2. What would happen to the surface gravity as the radius of the planet doubled?
If the radius of the planet doubled, the mass of the planet increases by a factor of 8 (2^3), assuming the density remains constant. Using the formula for gravitational acceleration, g = G * (M / r^2), we see that when the mass increases by a factor of 8 and the radius doubles, the force of gravity at the planet's surface will increase by a factor of 8 (2^3). This means that the surface gravity will also increase by a factor of 8.
In summary, the force of gravity at the surface of a growing planet increases as its mass and radius increase. The force of gravity is directly proportional to the mass and inversely proportional to the square of the radius. Therefore, as the mass and radius of the planet increase, the force of gravity at the surface also increases.