Problem 25 - College Physics

Question

An \(8.00 \mathrm{~kg}\) point mass and a \(15.0 \mathrm{~kg}\) point mass are held in place \(50.0 \mathrm{~cm}\) apart in deep space. They are both released at the same time. (a) Find the magnitude and direction of the acceleration of each particle the moment after they are released. (b) As the particles begin to move, is their acceleration constant? Explain.

Step-by-Step Solution

Answer

(a) The accelerations are \(4.0032 \times 10^{-9} \mathrm{m/s^2}\) and \(2.13504 \times 10^{-9} \mathrm{m/s^2}\). (b) No, the acceleration is not constant.

Step 1: Understanding the Problem

We have two point masses initially held in space: one with a mass of \(8.00\, \mathrm{kg}\) and the other with a mass of \(15.0\, \mathrm{kg}\), located \(50.0\, \mathrm{cm}\) apart. We need to find the initial accelerations and determine if the acceleration remains constant as they move.

Step 2: Calculating the Gravitational Force

The gravitational force between two point masses is given by \(F = G \frac{m_1 m_2}{r^2}\), where \(G = 6.674 \times 10^{-11}\, \mathrm{N m^2/kg^2}\), \(m_1 = 8.00 \, \mathrm{kg}\), \(m_2 = 15.0 \, \mathrm{kg}\), and \(r = 0.50 \, \mathrm{m}\). Substitute these values to find \(F\).

Step 3: Calculate Gravitational Force

Substitute into the formula: \[ F = \frac{(6.674 \times 10^{-11} \, \mathrm{N m^2/kg^2}) \times (8.00 \, \mathrm{kg}) \times (15.0 \, \mathrm{kg})}{(0.50 \, \mathrm{m})^2} \]Calculate to find \(F = 3.20256 \times 10^{-8} \, \mathrm{N}\).

Step 4: Finding the Acceleration of Each Mass

The acceleration is given by \(a = \frac{F}{m}\). For the \(8.00 \, \mathrm{kg}\) mass, \[ a_1 = \frac{3.20256 \times 10^{-8} \, \mathrm{N}}{8.00 \, \mathrm{kg}} \approx 4.0032 \times 10^{-9} \, \mathrm{m/s^2} \].For the \(15.0 \, \mathrm{kg}\) mass, \[ a_2 = \frac{3.20256 \times 10^{-8} \, \mathrm{N}}{15.0 \, \mathrm{kg}} \approx 2.13504 \times 10^{-9} \, \mathrm{m/s^2} \].

Step 5: Determine Direction Relating to Each Mass

Both masses will accelerate towards each other. The \(8.00 \, \mathrm{kg}\) mass accelerates towards the \(15.0 \, \mathrm{kg}\) mass and vice versa.

Step 6: Evaluating Constant Acceleration

As the masses move, \(r\) decreases leading to an increase in gravitational force, thus the acceleration changes. The acceleration is not constant because it depends on \( \frac{1}{r^2} \).

Key Concepts

Gravitational Force Calculation

Newton's Law of Universal Gravitation helps us calculate the force between two masses in space. The formula is essential:

• \[ F = G \frac{m_1 m_2}{r^2} \]
• Where:
  • \( F \) is the gravitational force,
  • \( G \) is the gravitational constant,
  • \( m_1 \) and \( m_2 \) are the masses of the objects,
  • \( r \) is the distance between the centers of the two masses.

Understanding these components is crucial when calculating gravitational force. Each value needs to be placed correctly to get accurate results. This calculation reveals how much pull one mass has on another purely due to gravity.

Acceleration of Point Masses

Acceleration originates from the gravitational force acting on an object. Each mass accelerates proportionally to the gravitational force applied to it, using the formula:

• For any given mass \( m \), the acceleration \( a \) is calculated as:

\[ a = \frac{F}{m} \]This value quantifies how rapidly the velocity of each mass changes and will differ for each mass based on its size:

• Heavier objects experience smaller acceleration.
• Lighter objects accelerate more for the same force.

By applying this concept, we learn that each point mass experiences different levels of acceleration towards each other, calculated separately.

Motion in Deep Space

In deep space, there are negligible external forces like friction or air resistance, allowing objects to demonstrate pure Newtonian motion. When two masses interact gravitationally:

• They move towards each other based solely on their mutual gravitational attraction.
• This relative motion is constrained by the initial separation distance and individual masses.

With no interference, the analysis assumes that gravitational forces are the primary influences. Consequently, these motions beautifully illustrate Newton's laws in their simplest form, reflecting perfect interactions as suggested by theory.

Variable Acceleration

The acceleration between two masses isn't constant due to the dependency on distance \( r \). As the masses move closer, \( r \) decreases, resulting in:

• Increased gravitational force.
• Increased acceleration.

This variable acceleration is described by:\[ a = G \frac{m_1 m_2}{r^2 m} \]Thus, for constant mass, only the change in \( r \) affects acceleration. This increasing acceleration as masses converge is a prime manifestation of law of universal gravitation in effect, showing real-time changes as distance decreases.

Force and Motion Analysis

Analyzing force and motion aims to understand how bodies move under the influence of forces. Motion is a response to these forces.

• Gravitational force serves as an initial push for movement.
• The motion that follows is greatly impacted by mass and initial distance.

Such analysis helps in:

• Predicting pathways.
• Determining speed over time.
• Understanding interactions in space correctly.

This concept clarifies how forces such as gravity fundamentally guide the evolution of motion, ensuring we comprehend how objects traverse through space.