Consider the problem of the particle moving on the surface of a cone. Show that the effective potential is V (r) = l2/(2 m r2) + mgr cot α (r is the radial distance in cylindrical coordinates, not spherical coordinates). Show that the turning points of the motion can be found from the solution of a cubic equation in r. Show further that only two of the roots are physically meaningful, so that the motion is confined to lie within two horizontal planes that cut the cone.
Consider the problem of the particle moving on the surface of a cone. Show that the effective potential is V (r) = l2/(2 m r2) + mgr cot α (r is the radial distance in cylindrical coordinates, not spherical coordinates). Show that the turning points of the motion can be found from the solution of a cubic equation in r. Show further that only two of the roots are physically meaningful, so that the motion is confined to lie within two horizontal planes that cut the cone.
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Consider the problem of the particle moving on the surface of a cone.
Show that the effective potential is
V (r) = l2/(2 m r2) + mgr cot α
(r is the radial distance in cylindrical coordinates, not spherical coordinates). Show that the turning points of the motion can be found from the solution of a cubic equation in r. Show further that only two of the roots are physically meaningful, so that the motion is confined to lie within two horizontal planes
that cut the cone.
Transcribed Image Text:### Equations of Motion in Generalized Coordinates
#### Diagram Explanation:
The image depicts a smooth cone with a half-angle denoted as α. Key components shown in the diagram include:
- **Cone**: Illustrated as a three-dimensional shape with a circular base and an apex at the origin.
- **Axes**: The coordinate system includes a vertical axis labeled as z and a horizontal plane with axes labeled r and θ.
- **Half-angle (α)**: The angle between the side of the cone and its axis.
- **Radius (r)**: The distance from the center of the base to a point on the edge of the cone.
- **Angle (θ)**: The angle in the horizontal plane from a reference direction.
#### Description:
7.4. A smooth cone of half-angle α.
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