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The centre of mass is a point in a body that acts as if the entire mass of the body is concentrated at that point. It is an essential part of mechanics. It helps perform calculations on massive bodies by assuming them as point objects. For instance, when we try to balance a rod by supporting it from a middle point, preventing it from falling. A lot of problems can be simplified if it is assumed that the total mass of an object is at one point. Generally, the centre of mass lies on planes of symmetry of bodies.
Determining the centre of mass
We should first start with simple shapes and try to find out their centre of mass. Most shapes like squares, triangles and circles have their centre of mass at their centroid. The centre of mass for bodies with a uniform and symmetrical shape is at their centroid. However, it is not that easy for bodies that do not have a symmetrical and uniform shape. For such types of bodies, their centre of mass could be anywhere.
Let us suppose a body that has a set of masses mi each at position ri then the centre of mass will be located at rcm which is given by
Mrcm = m1r1 + m2r2 + ….
rcm = = m1r1 + m2r2 + …./M
What is the centre of mass for two particles?
If we take two particles, one with mass m1 and the other with mass m2. Both of these particles lie on the x-axis. Particle 1 is at a distance of d1 from the centre O and particle 2 at d2. The centre of mass for these particles is at point C that lies at a distance D from point O. The equation will be:
D = m1d1 + m2d2 / m1 + m2
However, if we assume that the masses of each particle is the same then:
D = (d1 + d2) / 2
Determining the centre of mass questions
- A woman of 59 kg and a man of 73 kg are sitting on a seesaw that is 4.5 m long. Find the centre of mass.
Solution- m1 = 59 kg m2 = 73 kg
X1= 0 m x2 = 4.5 m
X com = (m1x1 + m2x2) / m1 + m2
X com = (59 * 0 + 73 * 4.5) / 59 + 73
X com = 328.5 / 132
X com = 2.49 m
- There are two bodies of mass 3 kg and 5 kg. They are located at 10 m and 5 m respectively. Find the centre of mass.
Solution – m1 = 3 kg m2 = 5 kg
X1 = 10 m X2 = 5 m
X com = (m1x1 + m2x2) / m1 + m2
X com = (3*10 + 5*5) / 3 + 5
X com = 55 / 15
X com = 3.67 m
- A 2000 kg truck moving at a velocity of 10m/s hits an 800 kg car. The car starts to move at 15m/s after the impact. It is assumed that the momentum is conserved during the collision, find out the velocity of the truck after the collision.
Solution- Before collision After collision
Truck = 2000 * 10 2000 * v
Car = 0 800 * 15
Total = 20000 20000
2000 * v + 12000 = 20000
2000 * v = 8000
V = 8000 / 2000
V = 4 m/s
Collisions
In space, when two freely moving particles get close to each other and for a short period of time they exert strong forces on each other until they move apart and start to move freely again, this phenomenon is known as Collision. The collision is said to be elastic if the mechanical energy is the same before and after the collision; else it is inelastic. However, if two objects collide with each other, stick together, and start to move as a combined mass then this is termed as a perfectly inelastic collision.
Conclusion
The centre of mass is a point in a body that acts as though all the mass of the body is concentrated on that point. It is of great importance and helps in describing the motion of the body. This point is said to be the weighted average of the positions of all the mass points. It is a concept that is applied in many areas such as gravitational science, electrostatistics, etc. By assuming the mass of an object at one point, it can help in simplifying many problems. The centre of gravity could be at different points depending on the symmetry of the objects.
