goldstein classical mechanics homework solutions

goldstein classical mechanics homework solutions

Goldstein Classical Mechanics Homework Solutions

1. Introduction

Classical mechanics is a subject that is about four hundred years old, beginning with the work of Galileo and Kepler, and later continued by Newton and many other scientists. That system had great success, enabling the theories developed at that time not only to describe the motion of many everyday situations as well as astronomical phenomena. With the development of atomic and sub-atomic physics at the beginning of the twentieth century, it was realized that classical mechanics could not account for the behavior of such small particles. And the success of quantum mechanics and relativity in the twentieth century has made classical mechanics seem incorrect and incomplete. However, again, if you look at the macroscopic world, which is the world we are living in and the world we all have direct experience with, it’s still the laws of classical mechanics invented by Newton that are in use. Obviously, in the realm of very small things such as collisions between gas molecules, or in the realm of very large things such as the motion of galaxies in the universe, we have to take into account the effects of relativity or quantum mechanics. But for the world we see, classical mechanics works. So in this subject, we are going to develop the classical theory carefully, starting from the very basic concepts and ending with the most sophisticated applications.

2. Kinematics

The expression in 2-17 has all the time dependence on the right hand side where it can be directly integrated. From 2-17, the velocity can be determined: v = v_o + ∫(t_o to t) a dt. Here, the limits of integration and the changing variables were used. This is true in all the cases where the acceleration only depends on the position. In each case, the integral automatically produces another variable that was never explicitly part of the sum. In 2-20, for example, each term of the sum could be integrated in dv and each term of the sum could be integrated in dx. Another example where it could be useful to employ different ways of expressing the derivatives and time derivatives of velocity and acceleration: from 2-20, dv/dt = d/dt(v_o + a_1 t + 1/2 a_2 t^2) = a_next + a.1. Where adoption of the notation in 2-17 could make calculation difficult and multiple different terms start, and multiple different terms start to get confusing. “next” is just a_1 (*1) and a indicates that the sum of the two terms is dv/dt, both written in a single intrinsic summand. These terms were already calculated by the time dv gets evaluated, so the answer is “ready written” on the line above (See this computation for an example). Overall, the notation provides a means to re-write derivatives of velocity or acceleration in different looking ways that are, in some cases, more useful for calculation than others. This could be the most useful feature of the notation described by 2-16. Such generality in notation can be helpful in deriving a more concrete answer based on the specific problem at hand. The author should have provided an example to illustrate this + give more meaning behind the introduction of the alternate notation presented over the course of this section. He could have also left a final concluding sentence on what this notation achieves in certain situations. Also a brief summarizing sentence for this section would be beneficial. Shutting down the section without a clear and re-cap in terms of the possibilities set by the alternate notation feels a bit anticipatory and out of place. (ChinYawen. M. on May 16, 2011)

3. Dynamics

Now, we will investigate the part of dynamics, which includes two sections. The main segment talks about the law of preservation of energy, which is one of the huge standards. It also talks about the time subordinate of energy. This gives us another method for taking care of issues in dynamics. Up until this point, Newton’s essential physical science has been covered generally in the course. For instance, kinematic conditions of uniform speeding and how Newton’s laws anticipate the powers in the event. In any case, it wasn’t until the twenty-first talk that power was communicated as the pace of progress of energy. This is the start of reasoning as far as dynamics. So it is not, at this point about portrayal of movement however perception of what it means for energy in the event. It will be fundamentally all the more fascinating when Newton’s laws are displayed in another light. This talked about in the subsequent themes; the change of the force ailment in light of the fact that the power is communicated as the rate of progress of energy. At that point we were acquainted with the idea of preservation of energy which is procured from information about work being never really body. At long last, the references to straight force and direct way were developed to turn out to be massively significant.

4. Energy and Momentum

The work finally proceeds to a discussion of momentum, a highly important principle in classical mechanics. Momentum and the closely related concept of energy are first considered as “scalar invariants of the motion of a particle.” That is, both energy and momentum are used to characterize the state of a particle, and being scalar invariants, the numerical values of the energy and momentum corresponding to a given state are the same for all observers that might be observing the particle in motion. This is an important point, for it means that energy and momentum measurements provide absolute knowledge of a particle’s state, which is not the case usually for velocity or force measurements, since the direction and the size of these vectoral quantities may change depending on the observer. However, momentum is also a “vectorial measure of motion.” This means that momentum, unlike energy, can be “stably associated with a particle as a kind of quality.” The direction of the momentum vector is the same as that of the velocity vector, and the size of the momentum vector is the product of the mass, m, of the particle and the size of the velocity vector. This also ties in to the origin of Newton’s second law. As a useful and powerful general property, the work demonstrates that if one takes the time derivatives of both sides of Newton’s second law, one arrives at vectorial expressions involving the time rate of change of velocity. This time rate of change, or derivative, on the left side of the equation is given a special name: acceleration. And so the second law provides a means to connect observed vectorial quantities, the force on a particle, with the derivative of the velocity vector, or acceleration. The work provides great insight as to why momentum is such a powerful concept by demonstrating that the total momentum of an isolated collection of particles is constant. By continuously improving and extending basic concepts, analyses and ideas, the work achieves great success towards the latter sections, by formulating the well-known conservation law: that the total momentum of an isolated collection of particles never changes with time. This development of momentum from simple linear momentum to angular momentum to Hamiltonian and Lagrange reformulations of particles in complicated systems to the conservation laws based on these formulations is argued in great mathematical detail.

5. Oscillations and Waves

The potential energy in the form of the harmonic oscillator is given by 1/2 – F(x) = 1/2 where F(x) = -kx. Here, k is the force constant and is a measure of the stiffness of the oscillator. For the simple pendulum, if we displace the bob by an angle ‘0 from its equilibrium position, the torque as a function of the angular displacement and its magnitude are given by ‘ = -mg. It is evident that the torque experienced by the pendulum is directly proportional to its angular displacement from the mean position, and tries to restore the pendulum to its equilibrium position. In the case of the conical pendulum, the bob moves in a circle on a horizontal plane with a constant speed. Due to the circular motion, there is always a net force acting toward the center of the circle (centripetal force). The radial component of tension T provides this centripetal force. By adding the parallel and perpendicular components of tension and using the small angle approximation, one can show that the angle ‘\beta between the string and the vertical is given by tan \beta = r(g/r). Note that the period of the conical pendulum is found to be T = 2\sqrt(g/r). When the mass is known, it can be related to the force constant and provides information about the stiffness. For a simple pendulum, a plot of the T^2/t(max)^2 on the y-axis versus the length on the x-axis will yield a linear relationship, which can be related to (so the slope) and g, the acceleration due to gravity. For a mass and spring system, we can apply a simple harmonic motion experiment and plot a graph of x versus t, and the period and force constant can be found. By using a computer program such as LoggerPro, it is not difficult to find the square of period and create a linear fit through the graph. Finally, for a conical pendulum, using an angle measuring instrument (such as a motion sensor), the angle can be given to find linear relationship and validate the theoretical predictions.