For which of the following does the centre of mass lie outside the body ?

The correct option is A bangle.

Understanding the Centre of Mass:
The centre of mass of a body is a unique point where the entire mass of the body can be assumed to be concentrated for analyzing its motion. The position of the centre of mass depends on the shape of the body and how its mass is distributed.

Crucially, the centre of mass is a geometric coordinate and does not necessarily have to lie within the physical material of the body.

Let us evaluate each of the given options:
- A pencil: Being a solid cylinder, its mass is distributed throughout its volume, and its centre of mass lies along its central axis inside the material.
- A shotput: As a solid sphere, its centre of mass lies at its geometric center, which is inside the solid metal.
- A dice: As a solid cube, its centre of mass lies at the geometric center, which is inside the solid material.
- A bangle: A bangle has the shape of a circular ring. The material and mass are distributed only along the outer circular rim, leaving the center completely empty. By symmetry, the centre of mass of a circular ring lies at its geometric center, which is in the hollow space. Thus, the centre of mass of a bangle lies outside its physical body.