91Ó°ÊÓ

The gravitational force is one of the fundamental forces in the universe. It pulls objects towards each other. Newton's law of universal gravitation helps us calculate this force between two masses. The formula is:

• \( F_g = \frac{G m_1 m_2}{r^2} \)

In this equation, \( F_g \) represents the gravitational force between two objects. \( G \), known as the gravitational constant, is a very tiny number \( 6.674 \times 10^{-11} \, \text{N} \, \text{(m/kg)}^2 \). This smallness shows how weak gravitational forces are unless the masses are very large.
\( m_1 \) and \( m_2 \) are the masses of the two objects exerting a gravitational pull on each other. \( r \) is the distance between the centers of these masses.
Even if the objects are far apart, gravitational force still exists. It's forever present, but weakens with distance squared \( r^2 \). This means increasing the distance between two objects will decrease the gravitational pull quickly.

On Earth, weight is what we call the force of gravity acting on an object. To find mass from weight, we use the formula:

• \( F = mg \)

Here, \( F \) is the force of weight measured in newtons \( (\text{N}) \), \( m \) is the object's mass in kilograms \( (\text{kg}) \), and \( g \) is the acceleration due to gravity on Earth, \( 9.81 \, \text{m/s}^2 \). To find mass:

• \( m = \frac{F}{g} \)

Let's take a look at our space probe example. For a weight \( F = 11000 \, \text{N} \), the mass \( m \) is calculated as:

• \( m = \frac{11000}{9.81} \approx 1121.30 \, \text{kg} \)

Similarly, with a weight of \( 3400 \, \text{N} \), we get:

• \( m = \frac{3400}{9.81} \approx 346.58 \, \text{kg} \)

Mass is a constant quantity and is the same regardless of location, whether on Earth or in space.

In space, the concept of distance becomes exceptionally important. As mentioned in Newton's law of universal gravitation, gravitational force decreases with an increase in the square of the distance \( r^2 \) between objects. Even minimal boost in distance separates objects noticeably in space.
Our exercise talks about a space probe with a center-to-center separation of \( 12 \, \text{m} \). This distance is essential in the calculation of the gravitational force.
In the formula \( F_g = \frac{G m_1 m_2}{r^2} \), as the value of \( r \) quadruples, the force \( F_g \) is reduced by a factor of \( 16 \) \((4^2)\). Therefore, understanding and measuring distance accurately is crucial in space calculations.
Moreover, distances in space are vast, so even small changes can lead to significant shifts in measurements and result outcomes.