Answer to Question #253455 in Classical Mechanics for Alwande

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 w about the z-axis.
d) A bead slides along a straight wire. The wire has constant inclination a
towards the horizontal x-y-plane and moves with constant acceleration in x
direction.

Answer 1

a)

Let the mass of the ball be m, radius of the ball be R , angular velocity be $\omega$

Angular momentum,

$v=r\times\omega$

So, it is holonomic constraint.

b)

It is an example of scleronomic constraint.

$\overrightarrow{r}=l\theta$

c)

As bead is freely can move along the rod and the rod is also rotating in xy plane,

So,

$(x-h)^2+(y-k)^2=l^2$

So, it is non-holonomic motion.

d)

Bead can move over along the straight wire,

$v=g\cos\theta t$

$\overrightarrow{r}=\frac{g\cos\theta t^2}{2}$

$r^2-a^2>0$

Hence, it is a non-holonomic constraint.