Question

An 8.00-kg point mass and a 12.0-kg point mass are held in place 50.0 cm apart. A particle of mass \(m\) is released from a point between the two masses 20.0 cm from the 8.00-kg mass along the line connecting the two fixed masses. Find the magnitude and direction of the acceleration of the particle.

Short answer

The acceleration is towards the 12.0-kg mass with a magnitude dependent on the net force.

Step-by-step solution

Understand the problem

We need to find the acceleration of a particle placed between two fixed masses. The forces acting on the particle are the gravitational attractions due to the two masses. Since there are two different forces, the net force on the particle will determine its acceleration.

Calculate the gravitational force from the 8.00-kg mass

Use Newton's Law of Universal Gravitation to determine the force exerted on the particle by the 8.00-kg mass. The formula is \[ F_1 = G \frac{m_1 m}{r_1^2} \]where \(m_1 = 8.00\, \text{kg}\), \(m\) is the particle's mass, \(r_1 = 0.20\, \text{m}\), and \(G = 6.67 \times 10^{-11} \text{N}\,\text{m}^2/\text{kg}^2\). This will give us the force acting towards the 8.00 kg mass.

Calculate the gravitational force from the 12.0-kg mass

Similarly, calculate the gravitational force exerted on the particle by the 12.0-kg mass using\[ F_2 = G \frac{m_2 m}{r_2^2} \]where \(m_2 = 12.00\, \text{kg}\), \(r_2 = (0.50 - 0.20)\, \text{m} = 0.30\,\text{m}\). This force acts towards the 12.0 kg mass.

Determine the net force on the particle

Since the two forces are in opposite directions, the net force \(F_{net}\) on the particle is the difference between \(F_2\) and \(F_1\), depending on which is greater. The particle will accelerate towards the mass with the stronger gravitational attraction.

Calculate the acceleration of the particle

Use Newton's second law of motion to win \[ a = \frac{F_{net}}{m} \] This equation gives the magnitude and direction of the acceleration of the particle, based on the net force and the particle's own mass.

Key concepts

Newton's Law of Universal Gravitation

Newton's Law of Universal Gravitation is pivotal for understanding the gravitational attraction between two masses. The law states that every point mass in the universe attracts every other point mass. The attractive force is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The formula, \ F = G \frac{m_1 m_2}{r^2} \, clearly highlights these relationships. Here, \(G\) is the gravitational constant, valued at approximately \(6.67 \times 10^{-11} \, \text{N}\,\text{m}^2/\text{kg}^2\). By understanding and using this law, we can calculate the gravitational forces between different bodies, such as the forces acting on a particle positioned between two point masses. This principle lays the foundation for further calculations, such as determining net forces and accelerations in gravitational scenarios.

Always remember that this force acts along the line joining the centers of the two bodies, meaning it will play a crucial role in scenarios such as our exercise.

Net Force Calculation

In physics, calculating the net force involves determining the sum of all forces acting on an object. When multiple forces are at play, like in the case of two different gravitational forces acting on a particle, it is essential to find out which force is stronger and the net effect.

The net force is what ultimately affects the motion of an object. For gravitational forces, if two masses exert forces in opposite directions on a third mass, the net force can be found by the difference in magnitude of these forces. Mathematically, it is expressed as \ F_{net} = F_2 - F_1 \, where \(F_2\) and \(F_1\) are the respective forces due to the two fixed masses.

In our exercise, once the individual gravitational forces are calculated using Newton's law, the net force will determine whether the particle accelerates towards the 8.00 kg mass or the 12.00 kg mass. This difference in forces allows us to calculate motion-related aspects like the particle's acceleration.

Newton's Second Law

One of the cornerstones of classical mechanics is Newton's Second Law of Motion. It provides the relationship between the net force acting on an object and its resulting acceleration. The law is succinctly captured in the equation \ F = ma \, where \(F\) is the net force, \(m\) is the mass of the object, and \(a\) is the acceleration.

This equation implies that the acceleration of an object is directly proportional to the net force acting upon it, while inversely proportional to its mass. Thus, with more force or less mass, the acceleration is greater, and vice versa.

In our specific problem, once the net force is calculated from the gravitational forces exerted by the two masses, applying Newton's Second Law helps determine the exact acceleration of the particle. By dividing the net force by the particle's mass, we obtain not only the magnitude but also the direction, indicating whether the motion is towards the 8.00 kg or the 12.00 kg mass.

Two-Body Problem

The two-body problem is a classic physics issue where the motion of two interacting bodies is studied. In our example, we have two fixed masses affecting a third particle. While the masses themselves do not move, their influence on the intermediary particle creates a dynamic problem.

The problem typically involves calculating the gravitational forces between the bodies and determining how these forces affect movement. This exercise is different from the usual orbit-based two-body problems because here, the fixed masses cause a particle to move through their gravitational attraction.

The solutions to two-body problems often use the principles of Newton's laws. Each body exerts a force on the other, causing a mutual influence that dictates their motion, which can be complex and requires thorough analysis. However, in this particular exercise, despite the simplicity of the setup, understanding the gravitational interaction remains central to determining the behavior of the particle.

Mass Interaction

Mass interaction occurs when two or more masses exert forces on each other. In physics, this typically involves gravitational forces, with each mass pulling on the other proportionate to its size and inversely proportional to the square of the distance between them.

This interaction is fundamental in many physical interactions and systems, from planetary orbits to simple exercises like the one we're addressing. Mass interaction dictates how masses affect not only one another directly, but also how a third mass like our particle is influenced by the surrounding masses.

In the given problem, the 8.00 kg and 12.00 kg masses are inducing forces on the particle due to their mass interaction via gravity. These forces lead to a net force, which then results in the particle accelerating. Understanding mass interaction is essential for mastering many physical concepts and is key to analyzing and predicting the outcomes of various physical scenarios.