###### Classical Mechanics

215 SOLVED PROBLEMS

Question: 6.8

## Finding 3-D potential energies (a) Show that the uniform gravity field F = −mgk is conservative with potential energy V = mgz. (b) Show that any force field of the form F = h(r)r (a central field) is conservative with potential energy V = −H(r), where H(r) ...

Since the potential energies are given, it is suff...
Question: 6.Q.2

## Does P leave the surface of the sphere? For the particular case in which u = (3gb)^1/2, show that P will leave the surface of the sphere, and find the value of θ at which it does so. ...

When $u=(3 g b)^{1 / 2}$, the formul...
Question: 8.3

## Proving existence of a limit cycle Prove that the autonomous system of ODEs x = x − y − (x² + y²)x, y = x + y − (x² + y²)y, has a limit cycle. ...

This system clearly has an equilibrium point at th...
Question: 8.4

## Rayleigh’s equation has a limit cycle Show that Rayleigh’s equation x + εx (x² − 1 ) + x = 0, has a limit cycle for any positive value of the parameter ε. ...

Rayleigh’s equation arose in his theory of the bow...
Question: 9.3

## Internal energy of three charged particles Three particles P1, P2, P3 carry electric charges e1, e2, e3 respectively. Find the internal potential energy Ψ. ...

In cgs/electrostatic units, the particles  ...
Question: 9.7

## Stability of a plank on a log A uniform thin rigid plank is placed on top of a rough circular log and can roll without slipping. Show that the equilibrium position, in which the plank rests symmetrically on top of the log, is stable. ...

Suppose that the plank is disturbed from its equil...
Question: 9.8

## Moment of inertia of a hoop Find the moment of inertia of a uniform hoop of mass M and radius a about its axis of rotational symmetry. ...

This is the easiest case to treat since each parti...
Question: 9.12

## Cylinder rolling down a plane A uniform hollow circular cylinder is rolling down a rough plane inclined at an angle α to the horizontal. Find the acceleration of the cylinder. ...

Suppose that, at time t, the cylinder has displace...
Question: 10.7