Problem 42 - University Physics with Modern Physics

Question

Four identical masses of 8.00 kg each are placed at the corners of a square whose side length is 2.00 m. What is the net gravitational force (magnitude and direction) on one of the masses, due to the other three?

Step-by-Step Solution

Answer

The net gravitational force is \( 140G\sqrt{2} \) directed diagonally.

Step 1: Understand the Problem

We need to find the net gravitational force on one mass, due to the other three masses placed at the corners of a square. The side length of the square is 2.00 m, and each mass is 8.00 kg.

Step 2: Identify the Forces

Each mass exerts a gravitational force on the chosen mass. There are three forces to consider: one from a mass directly adjacent, another adjacent at a right angle, and one diagonal.

Step 3: Calculate Gravitational Forces

Use Newton's Law of Gravitation. The force by a mass at distance \( r \) is given by \[ F = \frac{G \cdot m^2}{r^2} \] where \( G = 6.674 \times 10^{-11} \text{ N m}^2/\text{kg}^2 \), and \( m = 8.00 \text{ kg} \). For adjacent masses, \( r = 2.00 \text{ m} \). For the diagonal mass, \( r = \sqrt{2^2 + 2^2} = 2\sqrt{2} \).

Step 4: Calculate Forces for Adjacent Masses

For the adjacent masses,\[ F_{adj} = \frac{G \cdot 8^2}{2^2} = \frac{448 \cdot G}{4} = 112G \] This force acts along the sides of the square.

Step 5: Calculate Force for Diagonal Mass

For the diagonal mass,\[ F_{diag} = \frac{G \cdot 8^2}{(2\sqrt{2})^2} = \frac{448G}{8} = 56G \]This force acts diagonally.

Step 6: Determine Net Force Components

The two adjacent forces combine along the side directions (x and y), each contributing \(112G\) in magnitude. The diagonal force contributes to both axes, each component is \(\frac{56G}{\sqrt{2}} = 28G\). Thus, total net force components are\[ F_x = 112G + 28G = 140G \] and \[ F_y = 112G + 28G = 140G \].

Step 7: Calculate the Magnitude of the Net Force

Use Pythagorean theorem to find the magnitude of the net force:\[ F_{net} = \sqrt{F_x^2 + F_y^2} = \sqrt{(140G)^2 + (140G)^2} = 140G \sqrt{2} \].

Step 8: Determine the Direction of the Net Force

Given that the components are equal, the angle \( \theta \) is 45 degrees from either axis, so the net force acts diagonally.

Key Concepts

Newton's Law of Gravitation

Newton's Law of Gravitation is a fundamental principle in physics that describes how two bodies attract each other. The force of gravitational attraction between any two objects is given by the formula:\[F = \frac{G \cdot m_1 \cdot m_2}{r^2}\]where:

• \(F\) is the gravitational force between the objects,
• \(G\) is the gravitational constant, approximately \(6.674 \times 10^{-11} \text{ N m}^2/\text{kg}^2\),
• \(m_1\) and \(m_2\) are the masses of the objects,
• \(r\) is the distance between the centers of the two masses.

This law tells us that the gravitational force is directly proportional to the product of the masses and inversely proportional to the square of the distance between them. It helps us understand not only celestial motions but also interactions between everyday objects.

Mass and Distance Relationship

The relationship between mass, distance, and gravitational force is critical in understanding gravitational interactions. The force is larger when the masses involved are larger and decreases with increasing distance between the masses.

For two identical masses, like the ones described in the problem, the gravitational force diminishes as the distance \(r\) between the masses increases, according to the inverse-square law:

• If the distance between them doubles, the gravitational force is reduced to one-quarter.
• If the distance is halved, the force increases by a factor of four.

In the square configuration of the exercise, this relationship helps in calculating the net gravitational force by considering the differing distances between one mass and each of the others. Adjacent masses have a simple distance, while diagonal masses present a more complex, but calculable, distance.

Vector Components

In physics, understanding vector components is essential for calculating forces that are not aligned along a single axis. In this exercise, we deal with forces exerted in both the x and y directions, due to the square configuration.Each force exerted by the masses can be broken down into components along the x and y axes:

• Adjacent masses exert forces directly along one axis, either x or y.
• The diagonal mass exerts a force that can be split into x and y components using trigonometry. Since the sides of the square make 90-degree angles, the components of a diagonal force are equal: each is the original force divided by \(\sqrt{2}\).

By adding the contributions from both adjacent and diagonal components, we can determine the net force acting on the mass in question in the square setup.

Net Force Calculation

The net force calculation is the culmination of understanding the individual gravitational forces and their vector components. To find the net gravitational force on one mass due to the others, you add together the force components:1. Each adjacent mass contributes directly to either the x or y axis.2. The diagonal mass contributes equally to both axes.For each axis, you sum the contributions:

• \(F_x = 112G + 28G = 140G\)
• \(F_y = 112G + 28G = 140G\)

Finally, the magnitude of the net force is found using the Pythagorean theorem:\[F_{net} = \sqrt{F_x^2 + F_y^2} = 140G \sqrt{2}\]
The direction is analyzed, showing that it acts along a diagonal with an angle of 45 degrees due to equal components along x and y.