Answer to Question #145425 in Classical Mechanics for alwande hadebe

Question

Write down equations for the following constraints of mechanical systems
and classify them with respect to whether they are scleronomic or rheonomic and
holonomic or not nonholonomic.
a) A small ball which rolls down on the surface of a big ball without friction.
b) The suspension point of a planar pendulum carries out harmonic horizontal
oscillations.
c) A bead moving freely on a long straight wire through the origin, which rotates
in the x-y plane with constant angular velocity ω about the z-axis.
d) A bead slides along a straight wire. The wire has constant inclination α
towards the horizontal x-y-plane and moves with constant acceleration in x
direction.

Answer 1

a) A small ball which rolls down on the surface of a big ball without friction- It is non-holonomic and it's equation will be $r^2-a^2>0$

b)The suspension point of a planar pendulum carries out harmonic horizontal oscillations, it is an example of scleronomic system.$|\overrightarrow{r}^2|=l^2$

c)A bead moving freely on a long straight wire through the origin, which rotates

in the x-y plane with constant angular velocity ω about the z-axis. It is an example of a non-holonomic constraint.$z_1=z_2=0$

$(x_1-x_2)^2+(y_1-y_2)^2=l^2$

d)A bead slides along a straight wire. The wire has constant inclination α

towards the horizontal x-y-plane and moves with constant acceleration in x

direction. It is a example of non-holnomic. and it's equation will be $r^2-a^2>0$