Give the location of the centre of mass of a (i) sphere, (ii) cylinder, (iii) ring, and (iv) cube, each of uni

Exercises · 6.1

Q. Give the location of the centre of mass of a

(i) sphere,

(ii) cylinder,

(iii) ring, and

(iv) cube, each of uniform mass density. Does the centre of mass of a body necessarily lie inside the body?

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Concept understanding — Center of Mass

What is the Center of Mass?

Imagine you pick up a broom by the handle and try to balance it horizontally on one finger. You instinctively slide your finger along the handle until the broom stays level. That point — the one where the broom doesn't tip — is its center of mass.

Now think about throwing a cricket bat. It spins and wobbles in the air, but there is one point on the bat that follows a smooth, parabolic path, as if all the bat's mass were concentrated there. That point is also the center of mass.

The core idea is simple: the center of mass is the average position of all the mass in an object. It's the point where you could imagine the entire mass of the object being concentrated, and the object would behave the same way under the influence of external forces.

Why does this matter?

When you push an object at its center of mass, it moves in a straight line without rotating. Push it anywhere else, and it will both move and spin. This is why:

A car's stability depends on where its center of mass is (lower = safer).
A tightrope walker holds a long pole — moving the pole shifts their combined center of mass back over the rope.
In projectile motion, the center of mass of a system (like an exploding firework) continues along the original parabolic path, even though the fragments scatter.

The precise definition

For a system of particles, the center of mass is the weighted average of their positions, where the weight is the mass of each particle.

\vec{R}_{\text{CM}} = \frac{m_1 \vec{r}_1 + m_2 \vec{r}_2 + \cdots + m_n \vec{r}_n}{m_1 + m_2 + \cdots + m_n} = \frac{\sum m_i \vec{r}_i}{\sum m_i}

Here:

$\vec{R}_{\text{CM}}$ is the position vector of the center of mass
$m_i$ is the mass of the $i$ -th particle
$\vec{r}_i$ is the position vector of that particle

For a continuous object (like a rod or a sphere), the sum becomes an integral:

\vec{R}_{\text{CM}} = \frac{1}{M} \int \vec{r} \, dm

where $M$ is the total mass and $dm$ is an infinitesimal mass element.

Breaking it down with an example

Take two masses on a light rod: $m_1 = 2 \text{ kg}$ at $x = 0$ , and $m_2 = 3 \text{ kg}$ at $x = 5 \text{ m}$ .

The center of mass is:

x_{\text{CM}} = \frac{(2)(0) + (3)(5)}{2 + 3} = \frac{0 + 15}{5} = 3 \text{ m}

So the center of mass is at $x = 3 \text{ m}$ , closer to the heavier mass. That makes intuitive sense — the heavier mass "pulls" the average toward itself.

Note

The center of mass does not have to be inside the object. A ring or a hollow sphere has its center of mass at the geometric center, which is empty space.

Key properties to remember

External forces only — Internal forces (like collisions between parts of the system) do not affect the motion of the center of mass. Only external forces can change its velocity.
If no external force acts, the center of mass moves with constant velocity (or stays at rest). This is the law of conservation of momentum applied to the whole system.
For symmetric objects with uniform density, the center of mass coincides with the geometric center. For irregular shapes, it shifts toward the region with more mass.
In a uniform gravitational field, the center of mass and the center of gravity are the same point. (They differ only if gravity varies significantly across the object — not something you'll see in school problems.)

A final intuition

Think of the center of mass as the balance point of an object. If you could place a tiny, invisible support exactly at that point, the object would be perfectly balanced in any orientation. Every piece of mass on one side is exactly counterbalanced by the pieces on the other side.

That's why, when you jump off a boat, the boat moves backward — your center of mass and the boat's center of mass shift relative to each other, but the center of mass of the whole system (you + boat) stays put (if no external horizontal force acts). This is the heart of why the center of mass concept is so powerful: it lets you treat a complicated, spinning, wobbling object as a single point for many problems.

Looking up "Center of Mass: definition, formula & real-world examples" is a good habit before an exam, and it is worth knowing that Center of Mass is a core, NCERT-aligned topic from the System of Particles and Rotational Motion portion of the Class 11 Physics curriculum, and it is tested regularly in CBSE board exams as well as in JEE Main and NEET. Cross-checking this explanation against the relevant NCERT Physics chapter and solving a few past-year questions will round out your preparation.

For any uniform-density body, the centre of mass coincides with its geometric centre. For a sphere, cylinder, ring, and cube, these are respectively the sphere’s centre, the cylinder’s axis midpoint, the ring’s centre, and the cube’s centroid. The centre of mass need not lie inside the body — a ring is a classic example.

The centre of mass (COM) of a body is the point where the entire mass of the body can be considered to be concentrated for the purpose of analysing translational motion. For a body with uniform mass density, the COM coincides with the centroid of its shape — the geometric centre. This is because the mass is distributed symmetrically, so the average position of mass is exactly the shape’s centre.

Let’s locate the COM for each given shape.

Sphere (uniform density)

A sphere is perfectly symmetric about its centre in all three dimensions. Every point on one side has a mirror point on the opposite side with equal mass. The COM is therefore at the geometric centre — the centre of the sphere.
Cylinder (uniform density)

A solid cylinder has rotational symmetry about its axis and mirror symmetry about its mid-plane perpendicular to the axis. The COM lies on the axis, exactly halfway between the two flat faces — i.e., at the midpoint of the axis. For a cylinder of height $h$ , the COM is at a distance $h/2$ from either base, along the central axis.
Ring (uniform density)

A ring (a thin circular loop) has all its mass distributed along the circumference. By symmetry, the COM is at the centre of the ring — the point equidistant from all points on the ring. Notice that this point is not on the ring itself; it lies in the empty space inside the loop. This is a clear example that the COM can be outside the material of the body.
Cube (uniform density)

A cube is symmetric about its centre along all three axes. The COM is at the geometric centre — the point where the three diagonals intersect, i.e., at coordinates $(a/2, a/2, a/2)$ for a cube of side $a$ , taking one corner as the origin.

Watch out

A common mistake is to assume the centre of mass must lie inside the body. The ring shows this is false — the COM is at the centre of the empty hole. The COM is a mathematical point that can be anywhere in space, depending on mass distribution.

Now, to the second part: Does the centre of mass of a body necessarily lie inside the body?

No. The COM is the weighted average position of all mass elements. If the mass is distributed such that the average falls outside the material — as in a ring, a hollow sphere, or a horseshoe — the COM lies outside. The only requirement is that the COM lies on the line joining any two mass points, but it can be in empty space.

✓Final answer

The centre of mass is at the geometric centre for each: (i) sphere’s centre, (ii) cylinder’s axis midpoint, (iii) ring’s centre, (iv) cube’s centroid. The centre of mass does not necessarily lie inside the body — a ring is a counterexample.

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