Question

a) What is the total force on \(m_{1}\) due to \(m_{2}, m_{3},\) and \(m_{4}\) if all four masses are located at the corners of a square of side \(a\) ? Let \(m_{1}=m_{2}=m_{3}=m_{4}\). b) Sketch all the forces acting on \(m_{1}\).

Short answer

Answer: The total force on mass \(m_{1}\) can be found by calculating the gravitational forces due to each of the other three masses, and then summing them up as vector components along the \(x\) and \(y\) axes. The total force acting on \(m_{1}\) is given by: $$ \vec{F}_{1} = \sqrt{(\vec{F}_{1x})^2 + (\vec{F}_{1y})^2} $$ The sketch should include a square with the masses at its corners, arrows representing each force acting on \(m_{1}\), and the total force vector \(\vec{F}_{1}\) originating from \(m_{1}\) and acting in the direction of the vector sum of the forces. The sketch should also show the \(x\) and \(y\) components of the forces, demonstrating how the total force was calculated.

Step-by-step solution

Understand the gravitational force formula

Gravitational force between two masses (\(F\)) can be calculated using the formula: $$ F = G\frac{m1 * m2}{r^2} $$ where \(m1\) and \(m2\) are the two masses, \(r\) is the distance between the masses, and \(G\) is the gravitational constant.

Calculate the forces on \(m_{1}\) due to \(m_{2}, m_{3},\) and \(m_{4}\)

First, we need to calculate the forces acting on \(m_{1}\) due to each of the other masses separately, and then sum them up. The distance between \(m_{1}\) and \(m_{2}\) or \(m_{1}\) and \(m_{4}\) is \(a\). The distance between \(m_{1}\) and \(m_{3}\) is the diagonal of the square, which is \(\sqrt{2}\cdot{a}\). The force on \(m_{1}\) due to \(m_{2}\) is: $$ F_{12} = G\frac{m_{1} * m_{2}}{a^2} $$ The force on \(m_{1}\) due to \(m_{4}\) is: $$ F_{14} = G\frac{m_{1} * m_{4}}{a^2} $$ The force on \(m_{1}\) due to \(m_{3}\) is: $$ F_{13} = G\frac{m_{1} * m_{3}}{(\sqrt{2}\cdot{a})^2} $$

Sum up the gravitational forces acting on \(m_{1}\)

Next, we need to consider the direction of each force acting on \(m_{1}\). The total force on \(m_{1}\) can be represented by a vector sum: $$ \vec{F}_{1} = \vec{F}_{12} + \vec{F}_{13} + \vec{F}_{14} $$ The forces \(\vec{F}_{12}\) and \(\vec{F}_{14}\) are in the directions of the \(x\) and \(y\) axes, respectively, while \(\vec{F}_{13}\) acts along the diagonal. Now, we can break down \(\vec{F}_{13}\) into its \(x\) and \(y\) axis components: $$ \vec{F}_{13x} = F_{13} \cdot \cos(45^\circ) $$ $$ \vec{F}_{13y} = F_{13} \cdot \sin(45^\circ) $$ Finally, we can sum up the forces along the \(x\) and \(y\) axes: $$ \vec{F}_{1x} = \vec{F}_{12} + \vec{F}_{13x} $$ $$ \vec{F}_{1y} = \vec{F}_{14} + \vec{F}_{13y} $$

Calculate the total force on \(m_{1}\)

Now that we have the total force acting on \(m_{1}\) along the \(x\) and \(y\) axes, we can calculate the total force: $$ \vec{F}_{1} = \sqrt{(\vec{F}_{1x})^2 + (\vec{F}_{1y})^2} $$

Sketch all the forces acting on \(m_{1}\)

To sketch the forces, first draw a square with side \(a\) and place the masses at the corners. Then, draw arrows representing each force acting on \(m_{1}\), originating from the mass and pointing in the direction of the force: 1. \(\vec{F}_{12}\) acting horizontally from \(m_{1}\) towards \(m_{2}\). 2. \(\vec{F}_{14}\) acting vertically from \(m_{1}\) towards \(m_{4}\). 3. \(\vec{F}_{13}\) acting diagonally from \(m_{1}\) towards \(m_{3}\). Next, draw the \(x\) and \(y\) components of \(\vec{F}_{13}\), as well as the total forces \(\vec{F}_{1x}\) and \(\vec{F}_{1y}\) along the axes. Finally, draw the total force vector \(\vec{F}_{1}\), originating from \(m_{1}\) and acting in the direction of the vector sum of the forces. The sketch should clearly show the forces acting on \(m_{1}\) and their relationship to each other.

Key concepts

Newton's Law of Universal Gravitation

Understanding Newton's law of universal gravitation is fundamental when calculating forces in physics, particularly gravitational force. The law states that every particle attracts every other particle in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This can be expressed with the formula:

\[\begin{equation} F = G \frac{m1 \cdot m2}{r^2} \end{equation}\]
where F is the gravitational force, m1 and m2 are the masses of the two objects, r is the distance between the centers of the two masses, and G is the gravitational constant. Simplification of this exercise assumes equal masses, and we apply this law to calculate the gravitational force between them. The direct proportionality to the masses and inverse square relation to the distance are crucial in predicting how the force of gravity will act between two objects.

Vector Sum of Forces

When various forces act on an object, the vector sum of these forces determines the object's overall motion. For an object to be in equilibrium, for example, the vector sum of all forces acting upon it must be zero. This vector addition takes into account both magnitude and direction.

In our exercise, the forces on m1 due to m2, m3, and m4 are summed vectorially: \[\begin{equation} \vec{F}_{1} = \vec{F}_{12} + \vec{F}_{13} + \vec{F}_{14} \end{equation}\]
In practice, we often break forces into their x and y components, making the individual addition of these components much simpler before reconstructing them back into the resultant force vector. This method helps to visualize and calculate forces that are not aligned along the same axis.

Component of Force

Analyzing the component of a force is essential in understanding how it acts in different directions. Each vector force can be decomposed into parts that act along the axes of a coordinate system (usually x and y). This is particularly useful for forces acting at an angle, which is common in physics problems.

In our exercise, we decompose the diagonal force \vec{F}_{13} into its x and y components using trigonometric functions, assuming the angle to the x-axis as 45° due to the square symmetry: \[\[\begin{align*} \vec{F}_{13x} = F_{13} \cdot \cos(45^\circ) \vec{F}_{13y} = F_{13} \cdot \sin(45^\circ) \end{align*}\]\]
This technique simplifies our calculations because it allows us to add or subtract the forces separately along each axis. After computing the individual components, we can combine them to find the total force on an object in a particular direction.

Gravitational Constant

The gravitational constant, denoted by G, is a key value in Newton's law of universal gravitation. Its role is to provide the proportionality factor needed to make the units of measurement consistent and to ensure that the gravitational force equation yields force in units of Newtons. Currently, the accepted value is approximately \[\begin{equation} 6.674 \times 10^{-11} N(\frac{m^2}{kg^2}) \end{equation}\]
When using the gravitational force equation, this constant ensures that we are able to compare and calculate the gravitational attraction between objects, regardless of their mass or distance, in a standardized and reproducible manner. In the context of the exercise, G is fundamental to determining the specific force values between the masses located at the corners of a square.