Centre of Mass

Bodies have properties such as weight, charge, density, etc. Their weight is related to their mass and how this is affected by a gravitational force. In many cases, to simplify things, mass can be concentrated on a single point, especially when we are dealing with particles or small bodies. However, bodies can also be irregular or large, in which case we might need another means of simplification. It is here that the concept known as the centre of mass can be helpful.

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What is the centre of mass?

The centre of mass is the place where all the body’s mass is assumed to be concentrated. The concept of the centre of mass simplifies problems in two principal ways:

1. It provides a reference point for studying body-force interactions.
2. It simplifies the objects’ trajectories by representing their movement with the help of the centre of mass trajectory.

Forces

When using the centre of mass to study body-force interactions, the forces do not act at the contact point but rather on the centre of mass.

Figure 1. The centre of mass can help to simplify the calculation of forces acting on a body. In irregular bodies, the forces acting on the surface (left) can be simplified to forces acting on the centre of mass or ‘CM’ (right).

By concentrating all forces in the centre of mass, the laws of Newton and force superposition can be used to find the net force, the acceleration, and other variables.

Movement

If the forces acting on an object cause movement, this can be simplified to its centre of mass moving. In these cases, the centre of mass can be analysed using Newton’s laws on the centre of mass or the kinematic equations to obtain its displacement, velocity, and acceleration.

Figure 2. When a force F acts on a body, the movement can be represented as the movement of its centre of mass or ‘CM’. Here, the body moving from time T1 to time T2 follows a path prompted by the force ‘F’. The movement of the body can be represented by the black dot, which is its centre of mass.

Centre of mass of a square or rectangle

Some centres of mass can be determined more easily than others, depending on the object’s density, shape, and thickness. See the following example:

Let’s say you want to obtain the centre of mass of a regular and symmetrical body, such as a square. If you have ever played with a square piece of metal or wood, you know that you can balance it on your finger by placing it at the centre of the square.

Figure 3. The centre of mass in a square whose surface has a regular density is at its centre.

The balancing is possible because the square has a uniform density, which means that its weight is the same at every point. The force pulling it down (gravity) is also equal in all places.

The centre of mass for regular objects, such as a square, a rectangle, a circle, or an equilateral triangle, is at the centre of the geometrical shape, as shown below:

Figure 4. The centre of mass in a circle with an arbitrary radius r, an equilateral triangle with an arbitrary side l, a rectangle with arbitrary sides l and L, a ring with arbitrary inner and outer radii (r1 and r2).

For many regular figures, their centre of mass overlaps with a geometrical point known as their centroid.

The centroid is the centre of a geometrical shape.

Centroids

When an object’s density and shape are regular, the centre of mass is found at its geometrical centre or centroid, which is found in any regular 2D or 3D objects, such as spheres, cubes, and rings.

Figure 5. Centroids of regular 3D figures. If the density is regular, the centre of mass is the same as the centroid.

Centre of a system of particles

The centre of mass can be defined for a system consisting of several particles, such as when you analyse charges or point masses. If the objects have a regular density, the centre of mass can be found by using the following formula:

$CM = \frac{m_1\vec{r_1} +m_2\vec{r_2} +m_3\vec{r_3}}{m_1+m_2+m_3}$

Here, the vectors r are the coordinates x, y, and z measured from the origin. This is broken down in three formulas for the x, y, and z positions as follows:

$CM_x = \frac{m_1x +m_2x +m_3x}{m_1+m_2+m_3}$

$CM_y = \frac{m_1y +m_2y +m_3y}{m_1+m_2+m_3}$

$CM_z = \frac{m_1z +m_2z +m_3z}{m_1+m_2+m_3}$

Let’s look at an example to see how this works.

A system of three particles has the configuration shown in the image below. The particles are connected by forces that keep them locked in a triangular position. Another force makes all of them move along the y-axis.

Determine the centre of mass that can be used to simplify their movement if their individual masses are m1 =100 gr, m2=50 gr, and m3=64 gr.

Figure 6. System of particles with coordinates x, y, and z. The particles lie in the axis y at 0, so their second coordinate is 0.

First, you need to obtain the coordinate of each particle in the system. In this case, the particles lie along the position y=0. The problem will be simplified to find the coordinates in x and z.

$CM_x = \frac{m_1x +m_2x +m_3x}{m_1+m_2+m_3} = \frac{(100 \cdot 3) + (50 \cdot 2.5) + (64 \cdot 1.6)}{100+50+64} = 2.46 \approx 2.5$

$CM_z = \frac{m_1z +m_2z +m_3z}{m_1+m_2+m_3} = \frac{(100 \cdot 2.3) + (50 \cdot 3.5) + (64 \cdot 2.7)}{100+50+64} = 2.7$

You can then simplify the movement of the three particles as a single point in motion along the z and x-axes with the coordinates given below.

$CM_{x,z} = (2.5, 2.7)$

Figure 7. The centre of mass (pink) is the point that can be used to describe the trajectory of the three particles moving.

If the system consists of symmetrical objects with a uniform density, such as circles, squares, or rings, the coordinates of their centre of mass are provided by their centroids. Having obtained the centroid coordinates (x, y), these can be used to obtain the centre of mass of the whole system.

Centre of gravity

The centre of gravity is a useful concept, which helps us to simplify analyses of the forces acting on a single body or a system consisting of different bodies connected either physically or by a force.

The centre of gravity is the geometrical place where the force of gravity acts in a body or a system of bodies.

All physical objects have mass. If the mass is uniform, we can easily simplify the system of forces when we analyse a body in motion. In these cases, the whole mass can be placed in the centre of gravity, as the force of gravity acts on this point.

The centre of gravity also applies to bodies in rotation. In some of these cases, however, forces are not applied at the centre of mass but can cause torque and induce rotation. A classic example of the centre of mass in a body in rotation with regular density is a sphere.

A billiard ball is impacted exactly in the middle with force ‘F’. The ball can be considered a sphere with a regular density, and its centroid and centre of mass overlap. The ball is also affected by the force of gravity, which pulls it downwards. A third interaction that happens is ‘friction’.

Figure 8. The movement of the ball, the place where the force was applied, and gravity can be simplified as acting on the centre of mass (red x).

The force of friction ‘f’ acts parallel to the contact surface and against the movement of the ball. This force acts at a distance ‘d’ from the centre of the ball, causing a torque.

Figure 9. A billiard ball rotates due to the force of friction ‘f’, which is produced at a distance ‘d’ from its centre of mass after the ball was projected with a force F.

The torque caused by the friction is responsible for the ball’s rotation.

Centre of gravity vs centre of mass

The centre of gravity and the centre of mass should not be confused. One depends on the mass distribution of a body, while the other depends on the force of gravity that acts on the body. Both might overlap in special cases, such as when the mass of the object is regular and the gravitational field uniform. See the following example.

A bar with a length of 10 kilometres extends vertically over the earth’s surface. Its shape is cylindrical with a thickness A. The wind does not blow, and there is no other force other than gravity.

In this case, the centre of mass is exactly in the middle of the bar, as the density is regular. However, the centre of gravity is not. Let’s remember the formula for the force of gravity on the surface of the earth.

$F = G \cdot \frac{M_{Earth} \cdot m_{cylinder}}{r^2}$

Here, G is the earth’s gravity constant, which is 9.81 m/s2, r is the distance of the earth’s centre of mass from each part of the cylinder, which is measured in metres. MEarth and mcylinder are the mass of the objects (earth and each part of the cylinder), measured in kilograms.

The force decreases as we move further away from the earth’s surface. If you divide the bar into small cylinders with a 10cm height and a mass of 1kg, the last cylinders of that long bar feel less gravitational force than the ones on the surface if all values are the same, but only r is different.

This means that the top of the bar weighs less, and its centre of gravity moves downwards.

Some basic differences between the centre of mass and the centre of gravity are listed below.

 Centre of mass Centre of gravity Useful for analysing the movement of an object. Useful for analysing the stability of an object subjected to gravitational forces. Depends on the object’s mass. Depends on gravity. Can coincide with the centroid in symmetrical objects with regular density. May not coincide with the centroid in symmetrical objects with regular density because it depends on the position with respect to the field of gravity.

Projectile launch

During a projectile launch in parabolic motion, the centre of mass remains stable, even if the object rotates while travelling. In this case, the object rotates around its centre of mass, describing a parabola, as shown below.

Figure 10. An object being launched into a parabolic trajectory rotates around its centre of mass or ‘CM’, which describes a parabolic motion.

In the case of the launch of a rocket or a projectile, if the object separates into two parts, the motion of the centre of mass follows the original trajectory, while the parts of the projectile follow different paths, as shown below.

Figure 11. In a rocket launch, the separating parts of the rocket (a and b) follow different trajectories from that of the centre of mass (dotted line).

Centre of Mass - Key takeaways

• The centre of mass is the place where we can assume all mass to be concentrated in a body.
• The centre of mass is useful for simplifying the forces acting on a body or its motion.
• The centroid is the geometrical centre of an object.
• The centre of mass and the centroid coincide for bodies with regular symmetry and uniform density.
• The centre of gravity is the geometrical place where we can assume the force of gravity to be acting on a body.
• The centre of gravity and the centre of mass are not the same. One distinguishing feature is that the centre of gravity depends on the force of gravity, whereas the centre of mass does not.

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Frequently Asked Questions about Centre of Mass

Are the centre of gravity and centre of mass the same?

No, the centre of gravity is the geometrical position where the gravity force acts. In contrast to the centre of mass, it depends on the force of gravity.

Can the centre of mass change in a projectile launch?

No, the centre of mass will stay the same, even if the projectile itself divides, as in the case of rockets.

What is the centre of mass?

The centre of mass is the place where all the mass is assumed to be concentrated.

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What is the centre of mass?

How does the centre of mass help to simplify problems?

When forces act on a body, can we translate these forces to the centre of mass?

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