i) Classify each of the following according to whether they are (1) scleronomic or rheonomic, (2) holonomic or nonholonomic
a) A sphere rolls down from the top of a fixed sphere.
b) A cylinder rolls without slipping down a rough inclined plane.
c) A particle slides on a very long frictionless wire that rotates with constant angular velocity about a horizontal axis.
The correct answer and explanation is :
Classification and Explanation:
(a) A sphere rolls down from the top of a fixed sphere
- Scleronomic or Rheonomic:
- The constraints do not explicitly depend on time, making it scleronomic.
- Holonomic or Nonholonomic:
- The rolling condition introduces a velocity-dependent constraint (i.e., no slipping condition, ( v = R\omega )), which makes it nonholonomic.
(b) A cylinder rolls without slipping down a rough inclined plane
- Scleronomic or Rheonomic:
- The constraint conditions are independent of time (e.g., the equation of the inclined plane does not change with time), making it scleronomic.
- Holonomic or Nonholonomic:
- The no-slip condition (rolling constraint) is a velocity-dependent constraint that cannot be expressed solely as an equation in terms of coordinates and time. Thus, it is nonholonomic.
(c) A particle slides on a very long frictionless wire that rotates with constant angular velocity about a horizontal axis
- Scleronomic or Rheonomic:
- Since the wire rotates with constant angular velocity, its position explicitly depends on time. Therefore, the system is rheonomic.
- Holonomic or Nonholonomic:
- The motion of the particle is constrained to the wire and can be expressed in terms of its coordinates (e.g., cylindrical or polar coordinates). Since the constraints can be written purely in terms of coordinate equations, it is holonomic.
Explanation:
A scleronomic system has constraints that do not explicitly depend on time, whereas a rheonomic system has constraints that do. The first two cases involve constraints that do not change with time, making them scleronomic, while the third case has an explicitly time-dependent constraint due to the rotation of the wire.
A holonomic constraint can be written as an equation involving only the generalized coordinates and time, while a nonholonomic constraint involves inequalities or velocity-dependent constraints. Rolling motion without slipping introduces velocity-dependent conditions, making (a) and (b) nonholonomic. However, in (c), the motion is strictly along the wire and can be described by coordinate equations, making it holonomic.