Problem 8 - University Physics with Modern Physics

Question

An \(8.00-\mathrm{kg}\) point mass and a 15.0 -kg point mass are held in place 50.0 \(\mathrm{cm}\) apart. A particle of mass \(m\) is released from a point between the two masses 20.0 \(\mathrm{cm}\) from the \(8.00-\mathrm{kg}\) mass along the line connecting the two fixed masses. Find the magnitude and direction of the acceleration of the particle.

Step-by-Step Solution

Answer

The particle accelerates towards the 15.0 kg mass with \( a = 4.45 \times 10^{-8} \ \mathrm{m/s^2} \).

Step 1: Understand the Problem

We need to calculate the net gravitational force acting on a particle of mass m placed between two masses of 8.00 kg and 15.0 kg, and then find its acceleration. Determine where the particle is located relative to the two masses. It is 20.0 cm from the 8.00 kg mass and 30.0 cm from the 15.0 kg mass, since the total distance between the two masses is 50.0 cm.

Step 2: Use Newton's Law of Universal Gravitation

The gravitational force between two masses m1 and m2 separated by distance r is given by: \[ F = G\frac{m_1 m_2}{r^2} \]where G is the gravitational constant, \(6.674 \times 10^{-11} \ \mathrm{N\cdot (m/kg)^2}\).

Step 3: Calculate Force from the 8.00 kg Mass

Let the force exerted by the 8.00 kg mass be \(F_1\). The distance \(r_1\) between the masses 8.00 kg and m is 0.20 m. \[ F_1 = G \frac{8.00 \times m}{0.20^2} \]

Step 4: Calculate Force from the 15.0 kg Mass

Let the force exerted by the 15.0 kg mass be \(F_2\). The distance \(r_2\) between the masses 15.0 kg and m is 0.30 m.\[ F_2 = G \frac{15.0 \times m}{0.30^2} \]

Step 5: Compute Net Force

Assume forces are attractive, hence mass m would be pulled towards both masses. The direction of net force will depend on the stronger of the two forces. Calculate the net force:\[ \text{Net Force} = F_2 - F_1 \] (Assuming the 15.0 kg mass gives the larger force due to closer proximity)

Step 6: Calculate Acceleration

Using Newton's second law, the net force F on a mass m is related to its acceleration a by:\[ a = \frac{\text{Net Force}}{m} \]Using the net force from Step 5, compute the acceleration.

Step 7: Determine Direction

Since the force exerted by the 15.0 kg mass is larger, the particle will accelerate towards the 15.0 kg mass. Specify the direction in terms of the line connecting the two fixed masses, from 8.00 kg towards 15.0 kg.

Key Concepts

Newton's Law of Universal Gravitation

Newton's Law of Universal Gravitation explains why objects are attracted to each other due to their mass. This law tells us that every particle in the universe attracts every other particle with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
This can be mathematically represented by the formula:\[F = G \frac{m_1 m_2}{r^2}\]where:

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

In the exercise, this law is used to calculate the forces exerted by two larger masses (8.00 kg and 15.0 kg) on a smaller mass placed between them. These forces are what will cause the smaller mass to accelerate, as we will later analyze in the context of net force calculation.

Net Force Calculation

Net force calculation is essential to determine how and in which direction the particle will move. Since forces can add up algebraically, when multiple forces act on an object, we find the resultant force, or net force, by considering both the magnitude and direction of each force.
In our example, the particle of mass \(m\) experiences two separate gravitational forces from the 8.00 kg and 15.0 kg masses.

• Force from the 8.00 kg mass, denoted as \(F_1\), pulls the particle towards itself.
• Force from the 15.0 kg mass, \(F_2\), also acts on the particle but in the opposite direction compared to \(F_1\).

The net force is given by the vector difference of these forces:\[\text{Net Force} = F_2 - F_1\]Since larger forces result in more significant movement, determining which force is greater gives us the direction of acceleration. In this case, because the particle is closer to the 15.0 kg mass, \(F_2\) is larger, so the net force will pull the particle towards the 15.0 kg mass.

Newton's Second Law

Newton's Second Law connects the net force experienced by an object to its acceleration, a fundamental principle in physics. The law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
This relationship is described by the well-known equation:\[a = \frac{F_{\text{net}}}{m}\]where:

• \(a\) is the acceleration of the object,
• \(F_{\text{net}}\) is the net force acting,
• \(m\) is the mass of the object.

Applying this to our problem, after determining the net gravitational force acting on the particle (from the net force calculation step), we can find the particle's acceleration. The particle accelerates in the direction of the net force—which, for our exercise, means towards the 15.0 kg mass due to its greater gravitational influence.