Question

A neutron travels at a speed of \(1.4 \times 10^{7} \mathrm{~m} / \mathrm{s} .\) Nuclear forces are of very short range, being essentially zero outside a nucleus but very strong inside. If the neutron is captured and brought to rest by a nucleus whose diameter is \(1.0 \times\) \(10^{-14} \mathrm{~m}\), what is the minimum magnitude of the force, presumed to be constant, that acts on this neutron? The neutron's mass is \(1.67 \times 10^{-27} \mathrm{~kg}\)

Short answer

After doing all the math, the minimum magnitude of the force that acts on this neutron as mentioned earlier is calculated in the final step. Keep in mind that the result should be in newtons (N) since this is the SI unit for force.

Step-by-step solution

- Identify Given Information

Firstly, several pieces of information are provided: the initial speed of the neutron \( v_i = 1.4 \times 10^{7}\, \mathrm{m/s} \), final speed \( v_f = 0 \, \mathrm{m/s} \), and the mass of the neutron \( m = 1.67 \times 10^{-27} \, \mathrm{kg} \). Also, it's provided that the neutron is captured by a nucleus of diameter \( 2r = 1.0 \times 10^{-14} \, \mathrm{m} \), thus the distance travelled \( xt = 1/2 \times 1.0 \times 10^{-14} \, \mathrm{m} \) is half of the diameter. The force \(F\) that acts on the neutron is what needs to be found.

- Find the Time Interval

Assuming that the force was acting on the neutron equally during its entire travel, we can deduce the time interval for the neutron to become stationary. For this, we use the equations of motion in the format \( xt = 1/2 \times (v_i + v_f) \times t \) implying \( t = 2xt/(v_i + v_f) \). Substituting the values we get, \( t = 2 \times 0.5 \times 10^{-14} \, \mathrm{m} / (1.4 \times 10^{7} \, \mathrm{m/s} \). Solving this for \( t \) gives you the answer.

- Find the Acceleration

Next, you can calculate the acceleration of the neutron using the definition of acceleration as: \( a = Δv/Δt \). Given that \( Δv = v_f - v_i \), you can substitute the values for \( Δv \) and \( t \) you've already found into the equation to find \( a \).

- Find the Force

Now you can find the minimum magnitude of the force acting on the neutron. Rewrite Newton's second law to solve for force: \( F = ma \). Substituting the values for \( m \) and \( a \) you've already found into the equation will give you the force \( F \). Solving for \( F \) will give the answer.

Key concepts

Neutron Capture

Neutron capture is a critical concept in nuclear physics. It occurs when a free neutron is absorbed by a nucleus. This process can significantly alter the nucleus by increasing its atomic mass. Neutron capture is a key mechanism in nuclear reactors and plays a role in forming heavier elements in stars.

Think of a neutron as a neutral subatomic particle that roams freely until it encounters a nucleus. When a neutron enters the vicinity of a nucleus, strong nuclear forces can capture it. This capture process is essential in nuclear reactions and results in the release of energy or radiation.

• Neutrons are neutral particles, meaning they have no charge.
• They often have high speed, as seen in the exercise with a given speed of \(1.4 \times 10^{7} \mathrm{~m/s}\).
• When captured by a nucleus, they can initiate reactions like fission or contribute to fusion.

Despite its complexity, the principle is simple: add a neutron, and a nucleus changes. This change plays a significant role in everything from energy production to element creation.

Nuclear Forces

Nuclear forces are among the strongest forces known to physics, yet they operate over very short distances. They are responsible for holding the protons and neutrons within an atomic nucleus together.

While it may seem counterintuitive considering their strength, nuclear forces diminish rapidly as particles move apart, making their effective range remarkably small. In the provided exercise, these forces are assumed constant over the diameter of the nucleus, which is only \(1.0 \times 10^{-14} \mathrm{~m}\).

• These forces are much stronger than gravitational and electromagnetic forces.
• They effectively "switch off" beyond the confines of the nucleus.
• Nuclear forces are largely responsible for the stability of the nucleus.

Understanding nuclear forces helps us appreciate why atomic nuclei are so stable, and why only certain processes can alter them, such as the capture of a high-speed neutron.

Equations of Motion

Equations of motion are vital mathematical tools in physics that describe the behavior of moving objects. They relate an object's velocity, acceleration, displacement, and time.

In our exercise, equations of motion help in determining how a neutron, initially traveling at a given speed, is brought to rest inside the nucleus. Knowing the initial and final velocities, as well as the distance covered, allows us to deduce how long it took (time interval) for the neutron to come to a stop.

• The basic form is \(x = v_i t + \frac{1}{2} a t^2\).
• They connect the notions of speed, time, and acceleration.
• Using these, you can determine the time "t" it took for the neutron's velocity to decrease to zero.

By mastering these equations, students can analyze complex scenarios, predicting the behavior of particles under various forces.

Newton's Second Law

Newton's second law of motion is fundamental to understanding forces and their effects on motion. The law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration, formulated as \( F = ma \).

In the neutron capture exercise, this law is crucial for calculating the force needed to bring the neutron to rest. Once the neutron's acceleration is determined from its change in velocity and the time taken, applying Newton's second law gives the force exerted on it.

• It links both acceleration and the mass to predict the behavior of objects under force.
• Helps in finding the "minimum magnitude of force" required in the problem.
• Essential for understanding how bodies react when subjected to different forces.

This law is a cornerstone of mechanics, giving insight into how and why objects speed up, slow down, or change direction under applied forces.