Question

What is the magnitude of the free-fall acceleration of a ball (mass \(m\) ) due to the Earth's gravity at an altitude of \(2 R\), where \(R\) is the radius of the Earth?

Short answer

Answer: The magnitude of the free-fall acceleration is given by the expression \(a = G\frac{M}{(3R)^2}\), where \(G\) is the gravitational constant, \(M\) is the mass of the Earth, and \(R\) is the radius of the Earth.

Step-by-step solution

Gravitational force formula

We start by recalling the formula for the force between two objects due to gravity: \(F = G\frac{Mm}{r^2}\) where \(F\) is the gravitational force, \(G\) is the gravitational constant, \(M\) is the mass of the Earth, \(m\) is the mass of the ball, and \(r\) is the distance between the centers of the two objects. In this case, the distance \(r\) is equal to the sum of the Earth's radius and the altitude of the ball, so \(r = R + 2R = 3R\). We can now rewrite the formula as: \(F = G\frac{Mm}{(3R)^2}\)

Equate gravitational force to mass times acceleration

The gravitational force acting on the ball is also equal to its mass times its free-fall acceleration, which we'll denote as \(a\). So we can write: \(F = ma\)

Solve for acceleration

Now we can equate the expressions from steps 1 and 2, and solve for the acceleration: \(ma = G\frac{Mm}{(3R)^2}\) Divide both sides by the mass of the ball, \(m\): \(a = G\frac{M}{(3R)^2}\) That's the expression for the magnitude of the free-fall acceleration of the ball at an altitude of \(2R\).

Key concepts

Gravitational Force

Gravitational force is the attraction between two objects with mass. This force keeps planets in orbit and governs how objects fall on earth. Newton's law of universal gravitation quantifies this force. It says that any two objects attract each other with a force that is proportional to the product of their masses. This force is also inversely proportional to the square of the distance between their centers. The formula is \( F = G\frac{Mm}{r^2} \):

• \( F \) stands for gravitational force.
• \( G \) is the gravitational constant, a number that tells us about the strength of gravity. It's about \( 6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2 \).
• \( M \) and \( m \) are the masses of the objects.
• \( r \) is the distance between the centers of the two masses.

As the distance increases, the gravitational force becomes weaker. Thus, if you double the distance, the gravitational force is reduced to a quarter of the original strength.

Free-Fall Acceleration

Free-fall acceleration is the acceleration of an object caused solely by gravity. On Earth, this acceleration is denoted by \( g \) and is approximately \( 9.81 \, \text{m/s}^2 \) at the surface. When an object is in free-fall, it seems weightless because gravity is the only force acting on it. For an object at altitude, like the ball in the problem, the free-fall acceleration changes due to the increased distance to the center of the Earth.In the exercise, we found the expression for free-fall acceleration at an altitude of \( 2R \):

• At that distance, the acceleration \( a \) is \( G\frac{M}{(3R)^2} \).
• This formula helps us understand that acceleration reduces as we go further from the Earth's surface.

By plugging in the values for \( G \) and the Earth's mass \( M \), and calculating, you can find the acceleration value in \( \text{m/s}^2 \). This shows how gravity's pull becomes weaker with height.

Altitude

Altitude refers to the height above the Earth's surface. In the context of this exercise, altitude impacts how strongly gravity pulls on an object. Here's why altitude matters:

• The gravitational force decreases with increased altitude.
• Higher altitudes mean a larger distance \( r \) in the force equation \( F = G\frac{Mm}{r^2} \).
• If you are closer to the Earth's surface, gravity's pull is stronger.
• At \( 2R \) above Earth's surface where our ball is, the free-fall acceleration is not the same as \( g \).

Understanding altitude's impact on gravity helps in many areas like space travel, engineering, and even enjoying a day on a mountaintop. Altitude changes how forces interact, affecting how objects move and experience gravity.