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.