A planet has a mass M 1and radius R 1. The value of acceleration due to gravity on its surface is g 1. There is another planet 2, whose mass and radius both are two times that of the first planet. Which one of the following is the acceleration due to gravity on the surface of planet 2?
g 1/2
Understanding Acceleration Due to Gravity on a Planet
The question asks us to determine the acceleration due to gravity on the surface of a second planet compared to a first planet, given their relative masses and radii. The acceleration due to gravity on a planet's surface depends on its mass and radius.
Formula for Acceleration Due to Gravity
The acceleration due to gravity (\(g\)) on the surface of a spherical body with mass \(M\) and radius \(R\) is given by the formula:
\(g = \frac{GM}{R^2}\)
Where \(G\) is the universal gravitational constant.
Analyzing the First Planet
For the first planet, we are given:
- Mass = \(M_1\)
- Radius = \(R_1\)
- Acceleration due to gravity on the surface = \(g_1\)
Using the formula, the acceleration due to gravity on the surface of the first planet is:
\(g_1 = \frac{GM_1}{R_1^2}\)
Analyzing the Second Planet
For the second planet, we are given:
- Mass, \(M_2\), is two times the mass of the first planet: \(M_2 = 2M_1\)
- Radius, \(R_2\), is two times the radius of the first planet: \(R_2 = 2R_1\)
We want to find the acceleration due to gravity on the surface of the second planet, let's call it \(g_2\). Using the formula for \(g\):
\(g_2 = \frac{GM_2}{R_2^2}\)
Calculating Gravity on the Second Planet's Surface
Now, substitute the values of \(M_2\) and \(R_2\) in terms of \(M_1\) and \(R_1\) into the formula for \(g_2\):
\(g_2 = \frac{G(2M_1)}{(2R_1)^2}\)
Simplify the denominator:
\(g_2 = \frac{G(2M_1)}{4R_1^2}\)
Rearrange the terms:
\(g_2 = \frac{2G M_1}{4R_1^2}\)
Simplify the fraction \(\frac{2}{4}\) to \(\frac{1}{2}\):
\(g_2 = \frac{1}{2} \left(\frac{G M_1}{R_1^2}\right)\)
We know from the analysis of the first planet that \(g_1 = \frac{GM_1}{R_1^2}\). Substitute \(g_1\) into the equation for \(g_2\):
\(g_2 = \frac{1}{2} g_1\)
Thus, the acceleration due to gravity on the surface of planet 2 is half the acceleration due to gravity on the surface of planet 1.
| Property | Planet 1 | Planet 2 | Relation |
|---|---|---|---|
| Mass | \(M_1\) | \(M_2\) | \(M_2 = 2M_1\) |
| Radius | \(R_1\) | \(R_2\) | \(R_2 = 2R_1\) |
| Gravity | \(g_1\) | \(g_2\) | \(g_2 = \frac{G M_2}{R_2^2} = \frac{G (2M_1)}{(2R_1)^2} = \frac{2 G M_1}{4 R_1^2} = \frac{1}{2} \left(\frac{G M_1}{R_1^2}\right) = \frac{1}{2} g_1\) |
Conclusion
The acceleration due to gravity on the surface of planet 2 is \(g_1/2\).
Revision Table: Gravity Concepts
| Concept | Description | Formula |
|---|---|---|
| Acceleration due to Gravity (g) | The acceleration experienced by an object due to the gravitational force of a planet or star. | \(g = \frac{GM}{R^2}\) (on surface) |
| Universal Gravitational Constant (G) | A fundamental constant in the law of universal gravitation. Approx value \(6.674 \times 10^{-11} \text{ Nm}^2/\text{kg}^2\). | N/A (Constant) |
| Newton's Law of Universal Gravitation | Describes the gravitational force between two masses. | \(F = \frac{Gm_1 m_2}{r^2}\) |
Additional Information: Factors Affecting Acceleration Due to Gravity
The acceleration due to gravity on Earth (or any planet) is not perfectly uniform and can be affected by several factors:
- Altitude: As you move above the surface, the distance from the center of the planet (\(R\)) increases, causing \(g\) to decrease (since \(g \propto 1/R^2\)).
- Depth: Below the surface, the mass attracting an object decreases, and the formula becomes more complex. Gravity typically increases as you go down initially (reaching a maximum) and then decreases to zero at the center of the Earth.
- Shape of the Planet: Planets are not perfect spheres (Earth is an oblate spheroid, bulging at the equator). This means the radius is larger at the equator than at the poles, resulting in slightly lower gravity at the equator.
- Rotation of the Planet: The rotation of a planet creates a centrifugal effect that slightly counteracts gravity, especially near the equator, further reducing the effective acceleration due to gravity there.
- Local Density Variations: Variations in the density of the crust and mantle beneath the surface can cause small local differences in the acceleration due to gravity.
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