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You have successfully signed out and will be required to sign back in should you need to download more resources. Classical Mechanics, 3rd Edition. Safko, University of South Carolina. Overview Features Contents Order Overview. Description For thirty years this has been the acknowledged standard in advanced classical mechanics courses. The classic approach of this leading text has been revised and updated without increasing the overall length of the book.
Show that the components in the two coordinate systems are related to each other as in the equation shown below of generalized force 3. First lets find the components of the force in Cartesian coordinates. Convert U r, v into Cartesian and then plug the expression into the Lagrange-Euler equation. Thus the dot product simplifies and L is only the z-component. For part c, to obtain the equations of motion, we need to find the generalized kinetic energy.
With both derivations, the components derived from the generalized potential, and the components derived from kinetic energy, they will be set equal to each other. Find the generalized potential that will result in such a force, and from that the Lagrangian for the motion in a plane. Answer: This one takes some guess work and careful handling of signs.
To get from force to potential we will have to take a derivative of a likely potential. This has our third term we were looking for. Make this stay the same when you take the partial with respect to r.
Lets add to it what would make the first term of the force if you took the negative partial with respect to r, see if it works out. A nucleus, originally at rest, decays radioactively by emitting an electron of momentum 1. The MeV, million electron volt, is a unit of energy used in modern physics equal to 1.
In what direction does the nucleus recoil? If the mass of the residual nucleus is 3. The nucleus goes in the opposite direction of the vector that makes an angle 1. What are the equations of motion? What is the physical system described by the above Lagrangian?
Show that the usual Lagrangian for this system as defined by Eq. Derivation Answer: To find the equations of motion, use the Euler-Lagrange equations. If you make a substitution to go to a different coordinate system this is easier to see. Obtain the Lagrange equations of motion for spherical pendulum, i. When the rod is aligned along the z-axis, its potential will be its height. Find the equation of motion for x t and describe the physical nature of the system on the basis of this system.
Answer: I believe there are two errors in the 3rd edition version of this question. But we want to interpret it. So lets make it look like it has useful terms in it, like kinetic energy and force.
Two mass points of mass m1 and m2 are connected by a string passing through a hole in a smooth table so that m1 rests on the table surface and m2 hangs suspended. Assuming m2 moves only in a vertical line, what are the generalized coordinates for the system? Write the Lagrange equations for the system and, if possible, discuss the physical significance any of them might have. Reduce the problem to a single second-order differential equation and obtain a first integral of the equation.
What is its physical significance? Consider the motion only until m1 reaches the hole. The whole motion of the system can be described by just these coordinates. To write the Lagrangian, we will want the kinetic and potential energies. It is angular momentum. Now the Lagrangian can be put in terms of angular momentum. The next step is a nice one to notice.
If you take the derivative of our new Lagrangian you get our single second-order differential equation of motion. As far as interpreting this, I will venture to say the the Lagrangian is constant, the system is closed, the energy is conversed, the linear and angular momentum are conserved. Obtain the Lagrangian and equations of motion for the double pendulum illustrated in Fig 1. Answer: Add the Lagrangian of the first mass to the Lagrangian of the second mass.
Obtain the equation of motion for a particle falling vertically under the influence of gravity when frictional forces obtainable from a dissipation function 1 2 2 kv are present. A spring of rest length La no tension is connected to a support at one end and has a mass M attached at the other.
Neglect the mass of the spring, the dimension of the mass M , and assume that the motion is confined to a vertical plane. Also, assume that the spring only stretches without bending but it can swing in the plane.
Solve these equations fro small stretching and angular displacements. Solve the equations in part 1 to the next order in both stretching and angular displacement. This part is amenable to hand calculations. Using some reasonable assumptions about the spring constant, the mass, and the rest length, discuss the motion. Is a resonance likely under the assumptions stated in the problem? For analytic computer programs. To solve the next order, change variables to measure deviation from equilibrium.
Resonance is very unlikely with this system. The spring pendulum is known for its nonlinearity and studies in chaos theory.
A particle moves in the xy plane under the constraint that its velocity vector is always directed toward a point on the x axis whose abscissa is some given function of time f t. Show that for f t differentiable, but otherwise arbitrary, the constraint is nonholonomic. There can be no integrating factor for the constraint equation and thus it means this constraint is nonholonomic.
I will keep these two parts separate. The Z-axis adds more complexity to the problem. This switch makes sense because if you hang a rope from two points, its going to hang between the points with a droopy curve, and fall straight down after the points. This shaped revolved around the x-axis looks like a horizontal worm hole.
This is the classic catenary curve, or catenoid shape. The two shapes are physically equivalent, and take on different mathematical forms. Such equations of motion have interesting applications in chaos theory cf.
Chapter In analogy with the differential quantity, Goldstein Equation 2. This requires integration by parts twice. The first term vanishes once again, and we are still left with another integration by parts problem. Turn the crank again. Find the height at which the particle falls off.
The main purpose of this bibliography is to collect all the books given as "Further Reading" on individual theorem pages. Numbers in square brackets Computing at Columbia Timeline Books at Amazon.
The Amazon. Here you'll find current Relevant links posted in comments will be added. Click here for bottom P p p, P Momentum. Free download lesson plan, resume sample and terms paper in pdf. Choose from award-winning books and new releases.
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