John R. Taylor
ISBN #9781891389221
1st Edition
744 Questions
Homework Questions
Classical Mechanics is a comprehensive textbook that begins with the foundational principles of Newtonian motion, exploring themes such as conservation laws and the dynamics of particles under various forces. The text methodically develops the subject from basic projectile motion and energy conservation to advanced topics like the calculus of variations, Lagrange's and Hamiltonian formulations, and chaos theory. With key chapters dedicated to angular momentum, oscillatory motion, rigid body dynamics, and continuum mechanics, it bridges theoretical concepts with practical experimentation and real-world applications. The book not only provides a robust framework for understanding classical physics but also lays the groundwork for further exploration into more complex systems and interdisciplinary applications.
Chapter 1
Newton's Laws of Motion
Chapter 2
Projectiles and Charged Particle
Chapter 3
Momentum and Angular Momentum
Chapter 4
Energy
Chapter 5
Oscillations
Chapter 6
Calculus of Variations
Chapter 7
Lagrange's Equations
Chapter 8
Two-Body Central-Force Problems
Chapter 9
Mechanics in Noninertial Frames
Chapter 10
Potational Motion of Rigid Bodies
Chapter 11
Coupled Oscillators and Normal Modes
Chapter 12
Nonlinear Mechanics and Chaos
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Chapter 13
Hamiltonian Mechanics
Chapter 14
Collision Theory
Chapter 15
Special Relativity
Chapter 16
Continuum Mechanics
Problem 1
The shortest path between two points on a curved surface, such as the surface of a sphere, is called a geodesic. To find a geodesic, one has first to set up an integral that gives the length of a path on the surface in question. This will always be similar to the integral (6.2) but may be more complicated (depending on the nature of the surface) and may involve different coordinates than $x$ and $y$. To illustrate this, use spherical polar coordinates $(r, \theta, \phi)$ to show that the length of a path joining two points on a sphere of radius $R$ is $L=R \int_{\theta_{1}}^{\theta_{2}} \sqrt{1+\sin ^{2} \theta \phi^{\prime}(\theta)^{2}} d \theta$ if $\left(\theta_{1}, \phi_{1}\right)$ and $\left(\theta_{2}, \phi_{2}\right)$ specify the two points and we assume that the path is expressed as $\phi=\phi(\theta)$ (You will find how to minimize this length in Problem $6.16 .)$
Ajay Singhal Numerade Educator
Problem 2
Given the two vectors $\mathbf{b}=\hat{\mathbf{x}}+\hat{\mathbf{y}}$ and $\mathbf{c}=\hat{\mathbf{x}}+\hat{\mathbf{z}}$ find $\mathbf{b}+\mathbf{c}, 5 \mathbf{b}+2 \mathbf{c}, \mathbf{b} \cdot \mathbf{c},$ and $\mathbf{b} \times \mathbf{c}$
Deborah Israel Numerade Educator
Problem 3
Go over the steps from Equation (1.25) to (1.29) in the proof of conservation of momentum, but treat the case that $N=3$ and write out all the summations explicitly to be sure you understand the various manipulations.
Prabhat Tyagi Numerade Educator
Problem 4
Consider the well-known problem of a cart of mass $m$ moving along the $x$ axis attached to a spring (force constant $k$ ), whose other end is held fixed (Figure 5.2 ). If we ignore the mass of the spring (as we almost always do) then we know that the cart executes simple harmonic motion with angular frequency $\omega=\sqrt{k / m} .$ Using the Lagrangian approach, you can find the effect of the spring's mass $M,$ as follows: (a) Assuming that the spring is uniform and stretches uniformly, show that its kinetic energy is $\frac{1}{6} M \dot{x}^{2} .$ (As usual $x$ is the extension of the spring from its equilibrium length.) Write down the Lagrangian for the system of cart plus spring. (Note: The potential energy is still $\frac{1}{2} k x^{2}$.) (b) Write down the Lagrange equation and show that the cart still executes SHM but with angular frequency $\omega=\sqrt{k /(m+M / 3)} ;$ that is, the effect of the spring's mass $M$ is just to add $M / 3$ to the mass of the cart.
Problem 5
A plane, which is flying horizontally at a constant speed $v_{\mathrm{o}}$ and at a height $h$ above the sea, must drop a bundle of supplies to a castaway on a small raft. (a) Write down Newton's second law for the bundle as it falls from the plane, assuming you can neglect air resistance. Solve your equations to give the bundle's position in flight as a function of time $t$. (b) How far before the raft (measured horizontally) must the pilot drop the bundle if it is to hit the raft? What is this distance if $v_{\mathrm{o}}=50 \mathrm{m} / \mathrm{s}$ $h=100 \mathrm{m},$ and $g \approx 10 \mathrm{m} / \mathrm{s}^{2} ?(\mathrm{c})$ Within what interval of time $(\pm \Delta t)$ must the pilot drop the bundle if it is to land within $\pm 10$ m of the raft?
Supratim Pal Numerade Educator
Problem 6
In this problem you will prove the equation of motion (9.34) for a rotating frame using the Lagrangian approach. As usual, the Lagrangian method is in many ways easier than the Newtonian (except that it calls for some slightly tricky vector gymnastics), but is perhaps less insightful. Let $\delta$ be a noninertial frame rotating with constant angular velocity $\boldsymbol{\Omega}$ relative to the inertial frame $\boldsymbol{\delta}_{\mathrm{o}}$. Let both frames have the same origin, $O=O^{\prime} .$ (a) Find the Lagrangian $\mathcal{L}=T-U$ in terms of the coordinates r and $\dot{\mathbf{r}}$ of $\delta$. [Remember that you must first evaluate $T$ in the inertial frame. In this connection, recall that $\mathbf{v}_{\mathbf{o}}=\mathbf{v}+\mathbf{\Omega} \times \mathbf{r} . \mathbf{J}(\mathbf{b})$ Show that the three Lagrange equations reproduce (9.34) precisely.
Keshav Singh Numerade Educator
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