Classical Mechanics by Gregory, R. Douglas | 9780521534093 | 521534097 | Holooly

Mechanical Engineering

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) ...

Verified Answer:

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. ...

Verified Answer:

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. ...

Verified Answer:

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. ...

Verified Answer:

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. ...

Verified Answer:

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. ...

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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. ...

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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. ...

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Suppose that, at time t, the cylinder has displace...

Question: 10.7

Using the elastic scattering formulae In an experiment, particles of mass m and energy E are used to bombard stationary target particles of mass 2m. ...

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Q. The experimenters wish to select particles that...

Question: 11.2

Total moment of gravity forces A system S moves under uniform gravity. Show that the total moment of the gravity forces about any point is the same as if all the mass of S were concentrated at its centre of mass. ...

Verified Answer:

Without losing generality, let the point about whi...

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