Chapter 6, Topics In Dynamics Video Solutions, An Introduction to Mechanics

Problem 1

Oscillation of bead with gravitating masses A bead of mass $m$ slides without friction on a smooth rod along the $x$ axis. The rod is equidistant between two spheres of mass $M$. The spheres are located at $x=0, y=\pm a$ as shown, and attract the bead gravitationally.

Find the frequency of small oscillations of the bead about the origin.

Dominador Tan

Numerade Educator

02:06

Problem 2

Oscillation of a particle with two forces A particle of mass $m$ moves in one dimension along the positive $x$ axis. It is acted on by a constant force directed toward the origin with magnitude $B$, and an inverse-square law repulsive force with magnitude $A / x^{2}$.

What is the frequency of small oscillations about the equilibrium point $x_{0}$ ?

Penny Riley

Numerade Educator

01:50

Problem 3

Normal modes and symmetry Four identical masses $m$ are joined by three identical springs, of spring constant $k$, and are constrained to move on a line, as shown.

There is a high degree of symmetry in this problem, so that one can guess the normal mode motions by inspection, without a lengthy calculation. Once the relative amplitudes of the normal mode motions are known, the normal mode vibrational frequencies follow directly.

Anand Jangid

Numerade Educator

05:37

Problem 4

Bouncing ball* A ball drops to the floor and bounces, eventually coming to rest. Collisions between the ball and floor are inelastic; the speed after each collision is $e$ times the speed before the collision where $e<1$ (e is called the coefficient of restitution). If the speed just before the first bounce is $v_{0}$, find the time to come to rest.

Jonathon Brumley

Numerade Educator

03:07

Problem 5

Marble and superball A small ball of mass $m$ is placed on top of a "superball" of mass $M$, and the two balls are dropped to the floor from height $h$. How high does the small ball rise after the collision? Assume that collisions with the superball are elastic, and that $m \ll M .$ To help visualize the problem, assume that the balls are slightly separated when the superball hits the floor. (If you are surprised by the result, try demonstrating the problem with a marble and a superball.)

Khoobchandra Agrawal

Numerade Educator

02:03

Problem 6

Three car collision Cars $B$ and $C$ are at rest with their brakes off. Car $A$ plows into $B$ at high speed, pushing $B$ into $C$. If the collisions are completely inelastic, what fraction of the initial energy is dissipated when car $C$ is struck? The cars are identical initially.

Keshav Singh

Numerade Educator

04:03

Problem 7

Proton collision A proton makes a head-on collision with an unknown particle at rest. The proton rebounds straight back with $\frac{4}{9}$ of its initial kinetic energy.

Find the ratio of the mass of the unknown particle to the mass of the proton, assuming that the collision is elastic.

Rashmi Sinha

Numerade Educator

04:33

Problem 8

Collision of $m$ and $M$ A particle of mass $m$ and initial velocity $v_{0}$ collides elastically with a particle of unknown mass $M$ coming from the opposite direction as shown in the left-hand sketch on the next page. After the collision, $m$ has velocity $v_{0} / 2$ at right angles to the incident direction, and $M$ moves off in the direction shown in the sketch. Find the ratio $M / m$.

Susan Hallstrom

Numerade Educator

04:33

Problem 9

Collision of $m$ and $2 m$ Particle $A$ of mass $m$ has initial velocity $v_{0}$. After colliding with particle $B$ of mass $2 m$ initially at rest, the particles follow the paths shown in the right-hand sketch. Find $\theta$.

Susan Hallstrom

Numerade Educator

01:57

Problem 10

Nuclear reaction in the L system In the $L$ system, a particle of mass $m_{1}$ with kinetic energy $E_{1}$ strikes a particle of mass $m_{2}$ initially at rest. A nuclear reaction occurs, with the release of a particle of mass $m_{3}$ at angle $\theta$ and energy $E_{3}$, and a particle of mass $m_{4}$ at angle $\phi$ and energy $E_{4}$. (Angles are measured in $L$ from the incident line.) Neither $\phi$ or $E_{4}$ are measured.

Find an expression for the energy $Q$ released in the reaction in terms of the masses, energies $E_{1}$ and $E_{3}$, and the angle $\theta$

Penny Riley

Numerade Educator

09:55

Problem 11

Uranium fission In a nuclear reactor powerplant, a very slow ("thermal") neutron has high probability of reacting with the uranium- 235 in the fuel rods. The ${ }^{235} \mathrm{U}$ then fissions asymmetrically into a light fragment (most likely strontium ${ }^{97} \mathrm{Sr}$ ) and a heavy fragment (most likely xenon ${ }^{138} \mathrm{Xe}$ ), releasing energy $170 \mathrm{MeV}$.

The fission also produces a few fast neutrons. These neutrons are slowed to thermal speeds by collisions as they pass through a moderator, possibly helium, graphite, or even ordinary water. Once slowed, they can induce additional fission events, so that the process becomes a self-sustaining chain reaction.
(a) What are the energies of the two fragments (in MeV) immediately after fission? Neglect energy carried off by the fast neutrons.
(b) A $1 \mathrm{keV}$ fast neutron (relative mass 1 ) in a moderator collides elastically with a helium atom ${ }^{4} \mathrm{He}$ (relative mass 4 ) at rest. What is the maximum amount of energy the neutron can lose?

Khoobchandra Agrawal

Numerade Educator

01:31

Problem 12

Hydrogen fusion The Sun generates most of its energy from a chain of nuclear reactions, beginning with the fusion of the nuclei of two atoms of ordinary hydrogen ${ }^{1} \mathrm{H}$. The process in the Sun requires high densities, about 160 times the density of water on the Earth, and extremely high temperatures, about $1.5 \times 10^{7} \mathrm{~K}$. These conditions enable the nuclei to approach close enough against the repulsive Coulomb force to allow the nuclei to come within the range of the strong nuclear force.

Efforts on the Earth to generate energy from fusion have centered mainly on the reaction between the two heavier isotopes of hydrogen, deuterium ${ }^{2} \mathrm{D}$ (one proton, one neutron) and tritium ${ }^{3} \mathrm{~T}$ (one proton, two neutrons):
$$
{ }^{2} \mathrm{D}+{ }^{3} \mathrm{~T} \rightarrow{ }^{4} \mathrm{He}+{ }^{1} \mathrm{n}+17.6 \mathrm{MeV}
$$
The products are ${ }^{4} \mathrm{He}$ (relative mass 4$)$ and a neutron (relative mass 1 ).

What are the energies of the products, in MeV? The deuterium and tritium are essentially at rest.

Katie Mcalpine

Numerade Educator

07:04

Problem 13

Nuclear reaction of $\alpha$ -rays with lithium $^{*}$ A thin target of lithium is bombarded by helium nuclei ( $\alpha$ -rays) of energy $E_{0}$. The lithium nuclei are initially at rest in the target but are essentially unbound. When an $\alpha$ -ray enters a lithium nucleus, a nuclear reaction can occur in which the compound nucleus splits apart into a boron nucleus and a neutron. The collision is inelastic, and the final kinetic energy is less than $E_{0}$ by $2.8 \mathrm{MeV}$. The relative masses of the particles are: helium, mass $4 ;$ lithium, mass $7 ;$ boron, mass $10 ;$ neutron, mass $1 .$ The reaction can be symbolized
$$
{ }^{7} \mathrm{Li}+{ }^{4} \mathrm{He} \rightarrow{ }^{10} \mathrm{~B}+{ }^{1} \mathrm{n}-2.8 \mathrm{MeV}
$$
(a) What is $E_{0, \text { threshold, the minimum value of }} E_{0}$ for which neutrons can be produced?
(b) Show that if the incident energy falls in the range $E_{0 . \text { drreshold }}<E_{0}<E_{0, \text { thresbold }}+0.27 \mathrm{MeV}$, the neutrons ejected in the forward direction do not all have the same energy but must have either one or the other of two possible energies. (You can understand the origin of the two groups by looking at the reaction in the center of mass system.)

Lisa Tarman

Numerade Educator

05:28

Problem 14

Superball bouncing between walls A "superball" of mass $m$ bounces back and forth with speed $v$ between two parallel walls, as shown. The walls are initially separated by distance $l$. Gravity is neglected and the collisions are perfectly elastic.
(a) Find the time-average force $F$ on each wall.
(b) If one surface is slowly moved toward the other with speed $V \ll v$, the bounce rate will increase due to the shorter distance

Keshav Singh

Numerade Educator

02:36

Problem 15

Center of mass energy Show that the energy of two non-interacting particles with masses $M_{a}$ and $M_{b}$ can be written $E=E_{0}+E^{\prime}$ where $E_{0}=\frac{1}{2} M V^{2}$ is the energy of $\mathrm{c}$. of $\mathrm{m}$. motion, $E^{\prime}=\frac{1}{2} \mu V_{r}^{2}$ is the energy in the $C$ system $M=M_{a}+M_{b}, \mathbf{V}$ is the velocity of the $\mathrm{c} .$ of $\mathrm{m} .$ in the $C$ system, and $\mathbf{V}_{\mathbf{r}}$ is the relative velocity of the particles.

Saman Zulfiqar

Numerade Educator

09:19

Problem 16

Converting between $\mathrm{C}$ and $L$ systems* A particle of mass $m$ and velocity $v_{0}$ collides elastically with a particle of mass $M$ initially at rest and is scattered through angle $\Theta$ in the center of mass $C$ system.
(a) Find the final velocity of $m$ in the laboratory $L$ system.
(b) Find the fractional loss of kinetic energy of $m .$

David Morabito

Numerade Educator

02:09

Problem 17

Colliding balls Two balls, of mass $m$ and mass $2 m$, approach from perpendicular directions with identical speeds $v$ and collide. After the collision, the more massive ball moves with the same speed $v$ but downward, perpendicular to its original direction. The less massive ball moves with speed $U$ at an angle $\theta$ with respect to the horizontal. Assume that no external forces act during the collision.
(a) Calculate the final speed $U$ of the less massive ball and the angle $\bar{\theta}$
(b) Determine how much kinetic energy is lost or gained by the two balls during the collision. Is this collision elastic, inelastic, or superelastic?

Anand Jangid

Numerade Educator