Question 14.6: Classify the following systems according to whether or not t...
Classify the following systems according to whether or not they are scleronomic or rheonomic, holonomic or nonholonomic, and conservative or nonconservative:
(a) a sphere rolling downward without friction on a fixed sphere;
(b) a cylinder rolling down on a rough inclined plane (inclination angle α);
(c) a particle gliding on the rough inner surface of a rotation paraboloid; and
(d) a particle moving without friction along a very long bar. The bar rotates with the angular velocity ω in the vertical plane about a horizontal axis.
(a) Scleronomic, since the constraint is not an explicit function of time. Nonholonomic, since the rolling sphere leaves the fixed sphere. Conservative, since the gravitational force can be derived from a potential.
(b) Scleronomic, holonomic, nonconservative: The equation of the constraint represents either a line or a surface. Since the surface is rough, friction occurs. Therefore this system is not conservative.
(c) Scleronomic, holonomic, but not conservative, since the friction force does not result from a potential!
(d) Rheonomic: The constraint is an explicit function of time. Holonomic: The equation of the constraint is a straight line that contains the time explicitly; conservative.