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Chapter 3: Problem 4

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

Expert verified
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

01

- 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.
02

- 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.
03

- 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 \( a \).
04

- 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.

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Most popular questions from this chapter

An electron is projected horizontally at a speed of \(1.2 \times\) \(10^{7} \mathrm{~m} / \mathrm{s}\) into an electric field that exerts a constant vertical force of \(4.5 \times 10^{-16} \mathrm{~N}\) on it. The mass of the electron is \(9.11 \times 10^{-31} \mathrm{~kg} .\) Determine the vertical distance the electron is deflected during the time it has moved forward \(33 \mathrm{~mm}\) horizontally.

A certain particle has a weight of \(26.0 \mathrm{~N}\) at a point where the acceleration due to gravity is \(9.80 \mathrm{~m} / \mathrm{s}^{2} .(a)\) What are the weight and mass of the particle at a point where the acceleration due to gravity is \(4.60 \mathrm{~m} / \mathrm{s}^{2} ?(b)\) What are the weight and mass of the particle if it is moved to a point in space where the gravitational force is zero?

What are the weight in newtons and the mass in kilograms of (a) a 5.00-lb bag of sugar, (b) a 240-lb fullback, and \((c)\) a 1.80-ton car? (1 ton \(=2000\) lb. \()\)

A rocket and its payload have a total mass of \(51,000 \mathrm{~kg} .\) How large is the thrust of the rocket engine when \((a)\) the rocket is "hovering" over the launch pad, just after ignition, and (b) when the rocket is accelerating upward at \(18 \mathrm{~m} / \mathrm{s}^{2} ?\)

Two blocks, with masses \(m_{1}=4.6 \mathrm{~kg}\) and \(m_{2}=3.8 \mathrm{~kg}\), are connected by a light spring on a horizontal frictionless table. At a certain instant, when \(m_{2}\) has an acceleration \(a_{2}=\) \(2.6 \mathrm{~m} / \mathrm{s}^{2},(a)\) what is the force on \(m_{2}\) and \((b)\) what is the acceleration of \(m_{1}\) ?
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