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

Explanation:

The center of mass of an object is the point at which the distribution of mass is equal in all directions. For symmetrical objects with uniform density, the center of mass lies at the geometric center. However, for some objects, the center of mass can lie outside the physical body of the object.

A pencil is a uniformly dense object, so its center of mass lies within its body. Similarly, a dice and a shotput are also uniformly dense objects, so their centers of mass lie within their bodies.

However, a bangle is a circular ring with a hollow center. The mass is distributed around the circumference of the ring, so the center of mass lies at the geometric center of the ring, which is outside the physical material of the bangle.

∴ The correct answer is option 3: a bangle.