Question

A spherical planet far out in space has mass \(\mathrm{M}_{0}\) and diameter \(\mathrm{D}_{0}\). A particle of \(\mathrm{m}\) falling near the surface of this planet will experience an acceleration due to gravity which is equal to (A) \(\left[\left(\mathrm{GM}_{0}\right) /\left(\mathrm{D}_{\circ}^{2}\right)\right]\) (B) \(\left[\left(4 \mathrm{mGM}_{0}\right) /\left(\mathrm{D}_{0}^{2}\right)\right]\) (C) \(\left[\left(4 \mathrm{GM}_{0}\right) /\left(\mathrm{D}_{0}^{2}\right)\right]\) (D) \(\left[\left(\mathrm{GmM}_{0}\right) /\left(\mathrm{D}_{\circ}^{2}\right)\right]\)

Short answer

The acceleration due to gravity acting on a particle of mass m near the surface of a spherical planet with mass M0 and diameter D0 can be found using the gravitational force formula and simplification. The acceleration due to gravity (g) is given by: \[\boxed{g = \frac{4GM_0}{D_0^2}}\] This corresponds to option (C).

Step-by-step solution

Understand the gravitational force formula

The gravitational force between two objects is given by Newton's Law of Universal Gravitation: \[F = G \frac{m_1 m_2}{r^2}\] Where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between the centers of mass of the two objects.

Rewrite the formula to find the acceleration due to gravity

We can rewrite the gravitational force formula by solving for the acceleration due to gravity acting on the particle (g) instead of the gravitational force. Since F = m * g, we have: \[g = \frac{F}{m} = G \frac{m_1 m_2}{m r^2}\] We can now substitute the planet's mass (M0) and particle's mass (m) and the distance between their centers of mass (D0): \[g = G \frac{M_0 m}{r^2}\]

Calculate the distance between the centers of masses

Since the particle is near the surface of the planet, the distance between their centers of mass (r) is approximately equal to the radius of the planet. Since diameter D0 = 2 * radius, we can find the radius: \[r = \frac{D_0}{2}\]

Substitute the distance into the equation for acceleration due to gravity

Now, let's substitute the found distance (r) into the equation for acceleration due to gravity (g): \[g = G \frac{M_0 m}{\left(\frac{D_0}{2}\right)^2} = G \frac{M_0 m}{\frac{D_0^2}{4}}\]

Simplify the equation

By simplifying the equation, we can cancel out the mass m of the particle and get the expression for the acceleration due to gravity only in terms of the gravitational constant (G), mass of the planet (M0), and diameter of the planet (D0): \[g = \frac{4GM_0}{D_0^2}\]

Compare with the given options

Comparing our found expression for the acceleration due to gravity (g) with the given options, we see that it corresponds to option (C): \[\boxed{g = \left[\left(4GM_0\right)/\left(D_0^2\right)\right]}\]

Key concepts

Gravitational Force

Gravitational force is the attractive force that one mass exerts on another. This force is fundamental to the universe and governs everything from the falling of an apple to the motion of planets. The strength of the gravitational force between two objects depends on two factors: their masses and the distance between their centers. As the mass of either object increases or the distance between them decreases, the gravitational force becomes stronger. Understanding this concept is crucial for comprehending how objects interact in space and why celestial bodies remain in their orbit. Mathematically, the gravitational force can be described using Newton's law of universal gravitation.

Newton's Law of Universal Gravitation

Newton's law of universal gravitation states that every point mass attracts every other point mass in the universe with a force directed along the line joining the two points. The formula for calculating this force is given by:\[ F = G \frac{m_1 m_2}{r^2} \]Where:

• \( F \) is the gravitational force between the two masses.
• \( G \) is the gravitational constant.
• \( m_1 \) and \( m_2 \) are the masses of the objects.
• \( r \) is the distance between the centers of the two masses.

This law helps us understand how gravity works not only on Earth but throughout the universe. By knowing the masses and the distance between them, we can predict the gravitational attraction between two objects.

Gravitational Constant

The gravitational constant, symbolized as \( G \), is a key part of Newton's law of universal gravitation. It remains constant throughout the universe and has a value of approximately\[ 6.674 \times 10^{-11} \text{ Nm}^2/ ext{kg}^2 \]The gravitational constant allows us to calculate the gravitational force for any two objects if their masses and the distance between them are known. This constant is essential for accurately modeling planetary motions, satellite orbits, and even the behavior of galaxies. Without it, we wouldn't have the precise mathematical framework needed to understand gravitational interactions.

Mass and Diameter of Spheres

When discussing objects like planets, their mass and diameter are crucial parameters. The mass \( M_0 \) of a planet is a measure of the amount of matter it contains, which directly affects its gravitational pull. The diameter \( D_0 \) provides information about the size of the planet and helps determine the distance between the surface and the center, known as the radius.Since the diameter is twice the radius, it is used to calculate the gravitational force a planet exerts. By knowing a planet's mass and diameter, we can determine the acceleration due to gravity on its surface, as illustrated in the given exercise. In this example, understanding these properties allows us to simplify complex gravitational equations, giving us insights into how gravitational forces work.