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

Introduction / Context:
The centre of mass of a body is a point at which the entire mass of the body may be considered to be concentrated for translational motion analysis. For many simple solids, the centre of mass lies inside the material. However, for some hollow or ring-shaped bodies, the centre of mass can lie in empty space. Recognising such cases is helpful for understanding rotational motion, stability, and balancing of objects.

Given Data / Assumptions:
• We compare several everyday objects: a pen, a cricket ball, a ring, a book, and a solid cube. • For each object, we consider its standard, uniform mass distribution. • The question asks where the centre of mass lies with respect to the material of the object.

Concept / Approach:
For a uniform solid body, such as a cricket ball, book, or solid cube, the centre of mass lies at the geometric centre, which is inside the material. For a ring or thin circular hoop, the mass is distributed along a circle and there is essentially empty space in the middle. The geometric centre of this circle is a point in space that has no material but is still the location where the mass distribution balances in all directions. This point is the centre of mass and lies outside the material of the ring itself.

Step-by-Step Solution:
Step 1: Consider a cricket ball, book, fountain pen, and solid cube. Each is mostly solid and filled, so their centre of mass is somewhere inside the material. Step 2: For uniform objects, the centre of mass usually coincides with the geometric centre; for the ball, cube, and book, the geometric centre is clearly within the object. Step 3: A fountain pen has an irregular shape but its centre of mass will still lie somewhere within the physical body, provided mass distribution is typical. Step 4: For a ring or thin hoop, the mass lies around a circular path, leaving an empty region at the centre. Step 5: The centre of mass of a uniform ring lies at the geometric centre of the circle, i.e., in the empty space inside the ring. Step 6: Therefore, among the given options, only the ring has its centre of mass outside the material of the body.

Verification / Alternative check:
If you support a uniform ring at a single point on its circumference and let it hang freely, it will orient so that the lowest point on the ring lies directly below its centre of mass. Observations show that the centre of each circle (a point with no material) is vertically above the point of support, indicating that the centre of mass is at this geometric centre in empty space. No such behaviour is seen for solid objects like a cricket ball or cube, where the centre of mass is inside the material.

Why Other Options Are Wrong:
Option a (A fountain pen): Although its mass distribution may be uneven, the centre of mass lies somewhere inside the pen body. Option b (A cricket ball): Being nearly spherical and solid, its centre of mass lies at its geometric centre inside the ball. Option d (A book): A book is a solid rectangular body whose centre of mass lies inside, at the intersection of its diagonals. Option e (A solid cube): Like the book, the cube's centre of mass is inside the material, at the geometric centre.

Common Pitfalls:
Beginners sometimes think the centre of mass must always lie within the physical material, but that is not true for hollow objects like rings and thin shells. Other examples where the centre of mass may lie outside the material include boomerangs and certain bent rods. The key idea is that centre of mass depends on mass distribution, not merely on the shape boundary.

Final Answer:
Among the given options, the object whose centre of mass lies outside the material is A ring.