An Overview of the Centre of Mass of a Thin Rod

This article introduces you to the concepts of the centre of mass and centre of gravity of a thin uniform rod.

The centre of mass is the point where the entire mass of a system of particles or a rigid body is distributed uniformly. Hypothetically, it is the point where the entire mass of the body is concentrated; i.e., if a body is in motion, then the characteristics of the movement of the body can be represented by the motion of that single point (the centre of mass).

The centre of mass of the body need not lie within the body. It can also lie outside the body such as in a hollow cylinder or in a ring.

Centre of Mass of a System of Particles

Let us consider a system with an ‘n’ number of particles and having mass m1, m2, m3……mn. Let C be the point where the centre of mass is located. The point C lies at a position (X,Y,Z) from the origin ‘O’ of the reference frame used to observe the system in space. The position of the ith particle is ( xi, yi, zi ) 

Then, the centre of mass of a system is 

X = (m1 x1 + m2 x2 + ….. + mn xn)/(m1 + m2 + ….. + mn)

which can be written as 

X = ( ⅀mi xi )/ M

where M is the total mass of the system i.e., M = m1 + m2 + ….. + mn

Similarly,

Y = ( ⅀mi yi )/ M

Z = ( ⅀mi zi )/ M

Centre of Mass of a Thin Rod

To find the centre of mass of a thin uniform rod, we consider the distribution of mass that is continuous throughout the rod. Therefore, we consider the thin uniform rod as a system of large numbers of particles with mass ∆m1, ∆m2, ……

Therefore,

⅀∆mi  becomes ∫dm, which is equal to the total mass of the thin uniform rod (say M)

And ⅀∆mi xi becomes  ∫xdm

Since the rod is very thin as compared to its length, its cross-sectional area (i.e., thickness) is neglected. Hence, only the length is considered for calculation.

For a thin rod of length ‘L’ (taken to be along the x-axis), the mass per unit length is M/L. Hence, for an infinitesimal part dx, the mass dm = (M/L) dx

Therefore, the centre of mass of a rod is calculated as follows:

X =  (∫xdm)/ ∫dm

  =  {∫x (M/L) dx }/ (∫dm)    [ Since dm = (M/L) dx ]

  =  (M/L) (∫xdx)/ ( ∫dm)

Integrating x with respect to dx, from 0 to L (the entire length of the rod) we get L2/2 

And  ∫dm = M 

Therefore, the above equation becomes 

X = (M/L)(L2/2)/M 

X = L/2

i.e ., The centre of mass of a uniform rod is at the middle point of the rod 

The Motion of Centre of Mass

The centre of mass in vector form is given by

    R = (⅀mi ri)/ M  

where R and ri are the position vector of the centre of mass and the ith particle, respectively.

       MR = (⅀mi ri) = m1 r1 + m2 r2 + m3 r3 + ….. + mn rn

Differentiating both sides of the above equation with respect to time, we get

M(dR/dt) = m1(dr1/dt) + m2(dr2/dt) + m3(dr3/dt) + ….. + mn(drn/dt)

We know that (dr/dt) is the velocity  of the particles. Therefore,

        MV = m1v1 + m2v2 + m3v3 + ….. + mnvn 

Again, differentiating both sides of the above equation with respect to time, we get the accelerations as

        MA = m1a1 + m2a2 + m3a3 + ….. + mnan 

Newton’s second law of motion states that the forces acting on each particle of the system/ body is
F1= m1a1, F1= m2a2, F3= m3a3 and so on

Therefore, we get     

MA = F1 + F2 + F3 + …. + Fn

where F1 + F2 + F3 + …. + Fn  is the sum of all the forces acting on the system/ body 

Therefore, 

F1 + F2 + F3 + …. + Fn = Fext      (i.e. external force acting on it)

    I.e., MA = Fext

From the above equation, we get that the centre of mass of the rod or body moves as the mass of the entire body is concentrated on the centre of mass and behaves as all the external force is acted upon.

The Centre of Gravity of a Thin Uniform Rod

The centre of gravity of a thin uniform rod is the point where the torque acting on the rod due to the gravitational forces (migi) acting on it is 0. As we consider g to be uniform therefore, the centre of gravity of a thin uniform rod is the same point as the centre of mass of a rod. Therefore, the term centre of gravity of a thin uniform rod or a body is sometimes used instead of the centre of mass.

Since we know that the centre of mass of a uniform rod is at the midpoint of the rod, hence, the centre of gravity of a thin uniform rod is also at the midpoint. 

Conclusion

To describe motion, a body can be considered a single point in its centre of mass. The centre of mass of a thin rod can be found exactly at its mid-point (half of its length). The centre of gravity and the centre of mass are at the same point under uniform gravity (i.e., where g is constant).