Classical Mechanics

Herbert Goldstein, Charles P. Poole Jr., John L. Safko

Chapter 1 Survey of the Elementary Principles - all with Video Answers

Educators

Chapter Questions

01:28

Problem 1

Show that for a single particle with constant mass the equation of motion implies the following differential equation for the kinetic energy:$$\frac{d T}{d I}-F \cdot \mathbf{v}$$ while if the mass varies with time the corresponding equation is $$\frac{d(m T)}{d r}-\mathbf{F} \cdot \mathbf{p}$$.

Penny Riley

Numerade Educator

04:25

Problem 2

Phone that the magnitude $R$ of the prition vector for the conter of mass from an arbitary crigin is given By the equation

Abdul Vahid M

Numerade Educator

15:06

Problem 3

Suppose a system of two perticles is known to obey the equations of motion, Eqs. (1.22) and $(1.26) .$ From the equations of the motion of the individual particles shrew that the internal farces Beween particles satisfy both the weal: and the strong laws of action and reaction. The argument may be generalized to a system with arbitrary rumber of particles, the proving the converse of the arguments learling to Eqs. (1.22) $\operatorname{and}(1,26)$.

Susan Hallstrom

Numerade Educator

20:33

Problem 4

The equations of constraint for the tolling disk, Eqs. (1.39) . are special cases of $\mathrm{gcn}$. eral linen differential equations of constraint uf the form
$$\sum_{r=1}^{n} 8_{i}\left(x_{1} \ldots \ldots . \mathrm{s}_{n}\right) d x_{f}=0$$

Kumareshwaran Rathinasabapathy

Numerade Educator

01:35

Problem 5

Two wheels of mediums $a$ are mounted co the chols of a comimon axic of length $b$ such that the wheels rotate independently. The whale combination mils without slipping on a plane, Show that there are two socioeconomic equations of constraint.$\cos \theta d x+\sin \theta d y=0$
$\sin \theta d x-\cos f d x=\frac{1}{2} a(d \phi+d \phi)$
(where $\theta, \phi$, and $\phi$ ' have meanings similar to these in the problem of a single sertical desk, and $(x, y)$ are the coordinates of a potate on the axic midway between the two wheels) and one holonemic equation of constraint.$$\theta-C-\frac{a}{b}\left(\phi-\phi^{\prime}\right)$$ .the axic midway between the two wheels) and one holonemic equation of

Narayan Hari

Numerade Educator

04:58

Problem 6

A particle mones in the $x y$ plane under the constant that its velocity vector is a ways directed tonvands a point on the $x$ axis whose abscica is some grien function of time $f(t)$. Show that for $f(t)$ differentiable, but odsrwise arbitrary, the constraint is nonholonomic.

Ashwin Banarsee

Numerade Educator

01:04

Problem 7

Show that Lagrange's equations in the form of Eys. (1.53) can also be wnitch as
\[\frac{\partial \dot{T}}{\partial \dot{q}_{j}}-2 \frac{\partial T}{\partial q_{j}}-Q_{j}\]There are sometimes latwon as the Nielsem furts of the Lingrange equatiouns.

Tyler Moulton

Numerade Educator

08:40

Problem 8

If $L$ is a Lagrangian for a systern of $n$ tegrecs of frectom satisfying Lagrange's rguations, show hy direct substitution that Exercises
\[L^{\prime}=L+\frac{d F\left(q_{1} \ldots . . q_{n} . t\right)}{d r}\]
alse satisfics Lagringe's equations where $F$ is any arbitmry, but differentiable, function of its arguments.

Roman Frolov

Numerade Educator

02:51

Problem 8

A. Lxyratesiza for a particulate physical systern can be written as
\[I_{2}^{\prime}=\frac{m}{2}\left(a_{2}^{2}+2 b_{2} v+c j^{2}\right)-\frac{R}{2}\left(a_{3}^{2}+2 b_{2} v+c_{2}^{2}\right)\]where $a, b,$ and $c$ ase artuitriny constants but satycect to the condition that $b^{2}-a c \neq 0$ What are the cquations of motion? Examine particularly the rwo cases $a=0=c$ and $b=0, c=-a,$ What is the physical system described by the above I agrangian? Show that the useal Lagrangian for this system as defined by $\mathrm{Fq} \cdot(157)$ is nelated to $L^{\prime}$ by a point transformation (cf. Derivation 10 ). What is the significance of the comalition on the value of $b^{2}-a c^{7}$

Narayan Hari

Numerade Educator

03:18

Problem 9

The electromagnetic field is invariant under a GATT transformation of the scalar and vector potential given by $$\begin{array}{l}
A \rightarrow A+\nabla \forall(r, t) \\
\phi \rightarrow \phi-\frac{1}{c} \frac{\partial \psi}{\partial r}
\end{array}$$ .Where $j$ is arhitrary (but differcritiatic). What effect does this pauge trinsformation have on the Lagrangian of a particle mowing in the electromagnetic ficld? Is the twotion affocted?

Bruce Edelman

Numerade Educator

02:28

Problem 10

Ict $q_{1} \ldots \ldots, q_{n}$ be a set of independent gereralized coordinates for a sysiem of $n$ degrees of freedom, with a Legrangian $L(q, \dot{q}, t)$. Suppose we trarsform to another set of independent coordinates $x_{1}, \ldots, 3_{n}$ by mares of transformation equations $$q_{1}=q_{1}\left(\eta_{1} \ldots \ldots, x_{2}, r\right), \quad i=1, \ldots, n$$ .(Such a transformition is callod a pount transformarinn.) Show that if the Lagrangian function is expressed as a function of $s_{f}, s_{f},$ and $r$ through the equations of transformation, then $L$ satisfics Lagranse's equations with respect to the coordinates:In other wordis, the form of Lagrange's cquations is imariant under a point tramformation.

Penny Riley

Numerade Educator

01:37

Problem 11

Consider a uniform thin disk that rolls without slipping on a horizontal plane. A horizontal force is applied to the center of the disk and in a direction parallel to the plane of the clish
(a) Derive Lagrange's equations and find the centralized force.
(b) Discuss the motion if the force is not applicably parallel to the plane of the did:

Penny Riley

Numerade Educator

01:16

Problem 12

The escape wherein of a par tick on Earth is the minimum velocity required at Earth's surface in criber that the particle can escape from Earth's gravitational fiend Neglecting the resistance of the atmosphere, the systemic is conservative. From the contervation theorem for petential plus kinctic encry show that the recape velocity for Earth. tencring the preacree of the Moon, is $11.2 \mathrm{km} / \mathrm{s}$

Kratika Bhadauria

Numerade Educator

06:30

Problem 13

Rockets are propeiled by the thomentum reacticn of the exhums pases expellod trom the tail since these gases arise frum the reaction of the fuels craned in the rucket. the mass of the rocket is not constant. but decreases as the fuel is expended. Show that the cquatica of motion for a recket projected vertically upward in a tanifurm gravitational field, negiccting atroospheric friction. is
\[m \frac{d v}{d r}=-v^{\prime} \frac{d m}{d t}-m g\]
where $m$ is the mass of the rocket and $v^{\prime}$ is the velocity of the escaping gases relative to the rocket. Integrate this equation to obtain v as a function of $m$, assuming a content time rate of loss of mass. Show, for a ruckct scarting initially from rest, with $x^{\prime}$ cqual $w 2.1 \mathrm{m} / \mathrm{s}$ and a mass luss per second equal to $1 / 60$ th of the initial mass, that in crider to ichch the cecepe velucity the ratio of de weight of the furel to the weight of the empty rocket must be almost 3001

Khoobchandra Agrawal

Numerade Educator

03:47

Problem 14

Two poirts of mass $m$ are joined by a rigid weightless reed of length $l$, the center of Which is convitrained to thove on a circle of mediums $a$. Express the kinetic cherpy in BCA-ralized crevedinates.

Vipender Yadav

Numerade Educator

05:11

Problem 15

A point partacle moves in space under the influence of a force derivatile from a sener. alized petential of the form $$(1, \mathbf{r}, v)=V(r)+\sigma \cdot \mathbf{L}$$ .Where $\mathbf{r}$ is the radus vector from a fixcd point. L is the angular memertum abcut that point, and $\boldsymbol{c}$ is a fixcd rector in spece
(a) Find the comporents of the force on the particle in hoth Carterian and spherical poln coordinates, on the basis of Eq (1.58)
(b) Show that the components in the rwo ceentinate systems are related to esch cother as in $\mathrm{Eq}(1.49)$
(c) Obtain the equations of motion in spherical polar coordinates.

Averell Hause

Carnegie Mellon University

17:45

Problem 16

A particle moves in a plane under the influence of a force, acting toward a counter of force, whose lithographic is $$F-\frac{1}{r^{2}}\left(1-\frac{r^{2}-2 r}{c^{2}}\right)$$ where $r$ is the distance of the particle to the center of force. Find the eencralired potential that will roult in such a force, and frum that the Lagrargian for the motion in a plane fThe expression for $F$ nepresents the force between two charyo in Wrber's cloctrodynamics,

Paul A.

California State Polytechnic University, Pomona

01:17

Problem 17

A ruclow, originally at rest, decays ndioactively by emirting an electron of marnentum 1.73 MeVIc, and at right angles to the direction of the clactron a ncutrino with momentum 1. $.00 \mathrm{McV} / \mathrm{c}$. (The MeV, million clectron volt, is a urrit of energy used in modern phy ics, cquat to $1,60 \times 10^{-13}$ ). Correspondingly. MeV/c is a unit of linear momentum crunl to $5.34 \times 10^{-22} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}$. In what diroction does the $4 \mathrm{u}$ cleus recoil' What is its momertum in MeVIc? If the mass of the residual nucleus is $390 \times 10^{-25}$ 18 what is its kinctic energy, in clectron volts?

Kratika Bhadauria

Numerade Educator

View

Problem 19

Obtain the Lagrange equations of motion tor a spherical pendulum, l.c, a thaws point descended by a rigid weightless rod.

Victor Salazar

Numerade Educator

06:45

Problem 20

A particle of thaws $m$ moves in che dimension such that is has the Lagging isn
\[L-\frac{m^{2} x^{4}}{12}+m i^{2} V(x)-V_{2}(x)\]
where $V$ is scrinc diferentiahle function of $x$. I'ind the equation of motion for $x(t)$ and describe the pluyscel furniture of the system oc the basis of this equation.

Lucas Finney

Numerade Educator

01:23

Problem 21

Two mass spiciness of mess $m_{1}$ and $m_{2}$ are connoted by a string posing through a hole in a smooth table so dus $m$, rests or the table surface and $m_{2}$ hangs suspended. Assuming $m_{2}$ moves only in a vertical line, what are the generalized coordinates for the system? Write the Lagrange equations for the system and, if possible, discus the physical significance any of them might hive. Reduce the problem to a single second-erder differental equetion and ohcain a first integral of the couation. What is is physical significince? (Ccnsider the motion only antil $m$, reaches the hole.)

Raj Bala

Numerade Educator

04:01

Problem 22

Obsain the I.agrangian and cquations of motion for the clowble pendulum illustrated in Fig $1.4,$ where the lengths of the pendula are $l_{1}$ and $l_{2}$ with corresponding thases $m$ : and $m_{2}$

James Kiss

Numerade Educator

04:43

Problem 23

Obizin the cquation of monodic for a particle falliny vertical under the influence of previty when frictional forcen ohtsinable from a dissipation function $\frac{1}{2} \mathrm{Le}^{2}$ are percent. Integrate the equation to chtain the velocity as a function of time and show that the maximum possible velocity for a fall from rest is $x=m x / L$.

Michael Jacobsen

Numerade Educator

02:26

Problem 24

A spring of rest length $L_{0}$ (no tension) is corrected to a support at one end and has a mass $M$ ettached at the other. Neglect the mass of the spring. the dimension of the mass $M_{\text {. If }}$ and as the mention is confined to a vertical plane. Also, assume that the spring only surctices without bendmé but it can swing in the plane.
(a) Using the angular displacement of the mass from the vertical and the engih the the string has stretched from ils rest length (hanging with the mass $m$ ), find $L$ as erange's equaticrs.
(b) Solve these cquaticas for small stretching and stresular displacements
(c) Solve the cquations ia part (a) to the next crefer its both steciching and angular Uisplacement Thas part is amenable to hand calculations. Lising sonse reascrable
(d) (For analytic compuler proprams.) Consicler the spring to have 2 total triss
(e) (For numenced computer analysis.) Make sets of reaconsble asstimptions of the starts the part (a) and mabre a single plict of the row continuation as functions of
time.

Kajal Gautam

Numerade Educator